Citation: Letizia Fracchia, Jareer J. Banat, Massimo Cavallo, Chiara Ceresa, Ibrahim M. Banat. Potential therapeutic applications of microbial surface-active compounds[J]. AIMS Bioengineering, 2015, 2(3): 144-162. doi: 10.3934/bioeng.2015.3.144
| [1] | Chunyan Luo, Yuping Yu, Tingsong Du . Estimates of bounds on the weighted Simpson type inequality and their applications. AIMS Mathematics, 2020, 5(5): 4644-4661. doi: 10.3934/math.2020298 |
| [2] | Sabir Hussain, Javairiya Khalid, Yu Ming Chu . Some generalized fractional integral Simpson’s type inequalities with applications. AIMS Mathematics, 2020, 5(6): 5859-5883. doi: 10.3934/math.2020375 |
| [3] | Muhammad Tariq, Hijaz Ahmad, Soubhagya Kumar Sahoo, Artion Kashuri, Taher A. Nofal, Ching-Hsien Hsu . Inequalities of Simpson-Mercer-type including Atangana-Baleanu fractional operators and their applications. AIMS Mathematics, 2022, 7(8): 15159-15181. doi: 10.3934/math.2022831 |
| [4] | Xuexiao You, Fatih Hezenci, Hüseyin Budak, Hasan Kara . New Simpson type inequalities for twice differentiable functions via generalized fractional integrals. AIMS Mathematics, 2022, 7(3): 3959-3971. doi: 10.3934/math.2022218 |
| [5] | Maimoona Karim, Aliya Fahmi, Shahid Qaisar, Zafar Ullah, Ather Qayyum . New developments in fractional integral inequalities via convexity with applications. AIMS Mathematics, 2023, 8(7): 15950-15968. doi: 10.3934/math.2023814 |
| [6] | Saad Ihsan Butt, Artion Kashuri, Muhammad Umar, Adnan Aslam, Wei Gao . Hermite-Jensen-Mercer type inequalities via Ψ-Riemann-Liouville k-fractional integrals. AIMS Mathematics, 2020, 5(5): 5193-5220. doi: 10.3934/math.2020334 |
| [7] | Areej A. Almoneef, Abd-Allah Hyder, Fatih Hezenci, Hüseyin Budak . Simpson-type inequalities by means of tempered fractional integrals. AIMS Mathematics, 2023, 8(12): 29411-29423. doi: 10.3934/math.20231505 |
| [8] | Naila Mehreen, Matloob Anwar . Some inequalities via Ψ-Riemann-Liouville fractional integrals. AIMS Mathematics, 2019, 4(5): 1403-1415. doi: 10.3934/math.2019.5.1403 |
| [9] | Ghulam Farid, Saira Bano Akbar, Shafiq Ur Rehman, Josip Pečarić . Boundedness of fractional integral operators containing Mittag-Leffler functions via (s,m)-convexity. AIMS Mathematics, 2020, 5(2): 966-978. doi: 10.3934/math.2020067 |
| [10] | Yousaf Khurshid, Muhammad Adil Khan, Yu-Ming Chu . Conformable fractional integral inequalities for GG- and GA-convex functions. AIMS Mathematics, 2020, 5(5): 5012-5030. doi: 10.3934/math.2020322 |
The following inequality is named the Simpson's integral inequality:
| $ |16[φ(ε)+4φ(ε+ζ2)+φ(ζ)]−1ζ−εζ∫εφ(t)dt|≤‖φ(4)‖∞(ζ−ε)4, $ | (1.1) |
where $ \varphi:[\varepsilon, \zeta]\rightarrow\mathbb{R} $ is a four-order differentiable mapping on $ (\varepsilon, \zeta) $ and $ \|\varphi^{(4)}\|_{ \infty} = \sup_{t\in(\varepsilon, \zeta)}|\varphi^{(4)}(t)| < \infty $.
Considering the Simpson-type inequality via mappings of different classes, many results involving the ordinary integrals can be found in [1,3,4,5,18,19] and the references therein.
Definition 1.1. A set $ \Omega\subseteq\mathbb{R}^{n} $ is named as invex set with respect to the mapping $ \eta:\Omega\times\Omega\rightarrow\mathbb{R}^{n}, $ if
| $ ε+η(ζ,ε)∈Ω $ |
holds, for all $ \varepsilon, \zeta\in \Omega $ and $ \theta\in[0, 1]. $
Definition 1.2. A mapping $ \varphi:\Omega\rightarrow\mathbb{R} $ is called preinvex respecting $ \eta $, if
| $ φ(ε+η(ζ,ε))≤(1−θ)φ(ε)+θφ(ζ), $ |
holds for all $ \varepsilon, \zeta\in\Omega $ and $ \theta\in[0, 1]. $
The preinvex function is an important substantive generalization of the convex function. For the properties, applications, integral inequalities and other aspects of preinvex functions, see [5,7,8,9,10,11,12,13,14] and the references therein.
Definition 1.3. Let $ s\in(0, 1) $ and a mapping $ \varphi:\Omega\rightarrow\mathbb{R} $ is called $ s $-preinvex respecting $ \eta $, if
| $ φ(ε+η(ζ,ε))≤(1−θs)φ(ε)+θsφ(ζ), $ |
holds for all $ \varepsilon, \zeta\in\Omega $ and $ \theta\in[0, 1]. $
Fractional calculus has recently been the focus of mathematics and many related sciences. In addition to defining new integral-derivative operators and their properties, many researchers have achieved important results in applied mathematics, engineering and statistics. It is appropriate for the reader to review articles [2,15,16,17] for some recent studies. We end this section by reciting a well known $ \kappa $-fractional integral operators in the literature.
Definition 1.4. ([6]) Let $ \varphi\in L[\varepsilon, \zeta], $ the $ \kappa $-fractional integrals $ J_{\varepsilon^{+}}^{\alpha, \kappa} $ and $ J_{\zeta^{-}}^{\alpha, \kappa} $ of order $ \alpha > 0 $ are defined by
| $ Jα,κε+=1κΓκ(α)x∫ε(x−θ)ακ−1φ(θ)dθ,(0≤α<x<ζ) $ | (1.2) |
and
| $ Jα,κζ+=1κΓκ(α)ζ∫x(θ−x)ακ−1φ(θ)dθ,(0≤α<x<ζ), $ | (1.3) |
respectively, where $ \kappa > 0, $ and $ \Gamma_{\kappa} $ is the $ \kappa $-gamma function defined as $ \Gamma_{\kappa}(x): = \int\limits_{0}^{\infty} \theta^{x-1}e^{-\frac{\theta^{\kappa}}{\kappa}}d\theta, \quad \Re(x) > 0, $ with the properties $ \Gamma_{\kappa}(x+\kappa) = x\Gamma_{\kappa}(x) $ and $ \Gamma_{\kappa}(\kappa) = 1. $
This paper aims to obtain estimation type results of Simpson-type inequality related to $ \kappa $- fractional integral operators. Next, we derive the results with the boundedness of the derivative and with a Lipschitzian condition for the derivative of the considered mapping to derive integral inequalities with new bounds. Application of our results to random variables are also provided.
Throughout of this work, let $ \Omega\subseteq\mathbb{R} $ be an open subset respecting $ \eta:\Omega\times\Omega\rightarrow\mathbb{R}\setminus\{0\} $ and $ \varepsilon, \zeta\in\Omega $ with $ \varepsilon < \zeta. $
To prove our results, we obtain a new integral identity as following:
Lemma 2.1. Let $ \varphi :\Omega = [\varepsilon +\eta (\zeta, \varepsilon)]\rightarrow \mathbb{R} $ be a differentiable function on $ (\varepsilon +\eta (\zeta, \varepsilon)) $ with $ \eta (\zeta, \varepsilon) > 0. $ If $ \varphi ^{\prime }\in L[\varepsilon, \varepsilon +\eta (\zeta, \varepsilon)], \, \, n\geq 0 $ and $ \alpha, \kappa > 0, $ then the following identity holds:
| $ Ψ(ε,ζ;n,α,κ):=η(ζ,ε)2(n+1)[1∫0(2(1−θ)ακ−θακ3)φ′(ε+1−θn+1η(ζ,ε))dθ+1∫0(θακ−2(1−θ)ακ3)]φ′(ε+n+θn+1η(ζ,ε))dθ], $ | (2.1) |
where
| $ Ψ(ε,ζ;n,α,κ):=16[φ(ε)+φ(ε+η(ζ,ε))+2φ(ε+1n+1η(ζ,ε))+2φ(ε+nn+1η(ζ,ε))−Γκ(α+κ)(n+1)ακ6(η(ζ,ε))ακ[Jα,κ(ε)+φ(a+1n+1η(ζ,ε))]+Jα,κ(ε+η(ζ,ε))−φ(ε+nn+1η(ζ,ε))]−Γκ(α+κ)(n+1)ακ3(η(ζ,ε))ακ[Jα,κ(ε+nn+1η(ζ,ε))+φ(ε+η(ζ,ε))+Jα,κ(ε+1n+1η(ζ,ε))+φ(ε)]]. $ |
Proof. Integration by parts, one can have
| $ I1=1∫0(2(1−θ)ακ−θακ3)φ′(ε+1−θn+1η(ζ,ε))dθ=n+13η(ζ,ε)φ(ε)+2(n+1)3η(ζ,ε)φ(ε+1n+1η(ζ,ε))−α(n+1)ακ+13κ(η(ζ,ε))ακ+1ε+1n+1η(ζ,ε)∫ε(ε+1n+1η(ζ,ε)−x)ακ−1φ(x)dx−2α(n+1)ακ+13κ(η(ζ,ε))ακε+1n+1η(ζ,ε)∫ε(x−ε)ακ−1φ(x)dx $ |
and
| $ I2=1∫0(θακ−2(1−θ)ακ3)]φ′(ε+n+θn+1η(ζ,ε))dθ=n+13(η(ζ,ε))φ(ε+η(ζ,ε))+2(n+1)3η(ζ,ε)φ(ε+nn+1η(ζ,ε))−α(n+1)ακ+13κ(η(ζ,ε))ακ+1ε+η(ζ,ε)∫ε+nn+1η(ζ,ε)(x−(ε+nn+1η(ζ,ε)))ακ−1φ(x)dx−α(n+1)ακ+13κ(η(ζ,ε))ακ+1ε+η(ζ,ε)∫ε+nn+1η(ζ,ε)((ε+η(ζ,ε))−x)ακ−1φ(x)dx. $ |
By adding $ I_{1} $ and $ I_{2} $ and multiplying the both sides $ \frac{ \eta(\zeta, \varepsilon)}{2(n+1)} $, we can write
| $ I1+I2=16[φ(ε)+φ(ε+η(ζ,ε))+2φ(ε+1n+1η(ζ,ε))+2φ(ε+nn+1η(ζ,ε))]−α(n+1)ακ6κ(η(ζ,ε))ακ[ε+1n+1η(ζ,ε)∫ε(ε+1n+1η(ζ,ε))ακ−1φ(x)dx+ε+η(ζ,ε)∫ε+nn+1η(ζ,ε)(x−(ε+nn+1η(ζ,ε)))ακ−1φ(x)dx]−α(n+1)ακ3κ(η(ζ,ε))ακ[ε+nn+1η(ζ,ε)∫ε(ε+η(ζ,ε)−x)ακ−1φ(x)dx+ε+1n+1η(ζ,ε)∫ε(x−ε)ακ−1φ(x)dx]. $ |
Using the fact that
| $ 1κΓκ(α)ε+1n+1η(ζ,ε)∫ε(x−ε)ακ−1φ(x)dx=Jα,κ(ε+1n+1η(ζ,ε))−φ(ε),1κΓκ(α)ε+η(ζ,ε)∫ε+nn+1η(ζ,ε)(ε+η(ζ,ε)−x)ακ−1φ(x)dx=Jα,κ(ε+nn+1η(ζ,ε))+φ(ε+η(ζ,ε)),1κΓκ(α)ε+1n+1η(ζ,ε)∫ε(ε+1n+1η(ζ,ε)−x)ακ−1φ(x)dx=Jα,κ(ε)+φ(ε+1n+1η(ζ,ε)),1κΓκ(α)ε+η(ζ,ε)∫ε+nn+1η(ζ,ε)(x−(ε+1n+1η(ζ,ε)))ακ−1φ(x)dx=Jα,κ(ε+η(ζ,ε))−φ(ε+nn+1η(ζ,ε)), $ |
we get the result.
Remark 2.1. If we choose $ \eta(\zeta, \varepsilon) = \zeta-\varepsilon $ with $ \kappa = 1 $, then under the assumption of Lemma 2.1 one has Lemma 2.1 in [19].
Theorem 2.2. Let $ \varphi :\Omega = [\varepsilon, \varepsilon +\eta (\zeta, \varepsilon)]\rightarrow \mathbb{R} $ be a differentiable function on $ \Omega. $ If $ \varphi ^{\prime }\in L[\varepsilon, \varepsilon +\eta (\zeta, \varepsilon)] $and $ |\varphi ^{\prime }|^{\lambda } $ for $ \lambda > 1 $ with $ \mu ^{-1}+\lambda ^{-1} = 1 $ is $ s $-preinvex function, then the following inequality holds for fractional integrals with $ \alpha, \kappa > 0, $ then the following integral inequality holds:
| $ |Ψ(ε,ζ;n,α,κ)|≤η(ζ,ε)2(n+1)(1∫0|2(1−θ)ακ−θακ3|μdθ)1μ×[[(1−1(n+1)s(s+1)|φ′(ε)|λ+1(n+1)s(s+1)|φ′(ζ)|λ)]1λ+[1(n+1)s(s+1)|φ′(ε)|λ+(21−s−1(n+1)s(s+1))|φ′(ζ)|λ]1λ]. $ | (2.2) |
Proof. From the integral identity given in Lemma 2.1, the H$ \ddot{o} $lder integral inequality and the $ s $-preinvexity of $ |\varphi ^{\prime }(x)|^{\lambda }, $ we have
| $ |Ψ(ε,ζ;n,α,κ)|≤η(ζ,ε)2(n+1)[1∫0|(2(1−θ)ακ−θακ3)||φ′(ε+1−θn+1η(ζ,ε))|dθ+|1∫0(θακ−2(1−θ)ακ3)||φ′(ε+n+θn+1η(ζ,ε))|dθ]≤η(ζ,ε)2(n+1)[(1∫0|(2(1−θ)ακ−θακ3)|μdθ)1μ×[1∫0((1−(1−θn+1)s)|φ′(ε)|λ+(1−θn+1)s|φ′(ζ)|λ)dθ]1λ+(1∫0|(θακ−2(1−θ)ακ3)|μdθ)1μ[1∫0((1−(n+θn+1)s)|φ′(ε)|λ+(n+θn+1)s|φ′(ζ)|λ)dθ]1λ]. $ | (2.3) |
Using the fact that $ 1-\chi ^{s}\leq (1-\chi)^{s}\leq 2^{1-s}-\chi ^{s} $ for $ \chi \in \lbrack 0, 1] $ with $ s\in (0, 1], $ we have
| $ 1∫0(1−(n+θn+1)s|φ′(ε)|λ+(n+θn+1)|φ′(ζ)|λ)dθ≤1∫0[(1−θn+1)|φ′(ε)|λ+(21−s−(1−θn+1)s)|φ′(ζ)|λ]dθ=1(n+1)s(s+1)|φ′(ε)|λ+(21−s−1(n+1)s(s+1))|φ′(ζ)|λ $ | (2.4) |
and
| $ 1∫0[(1−(1−θn+1)s)|φ′(ε)|λ+(1−θn+1)s|φ′(ζ)|λ]dθ=(1−1(n+1)s(s+1)|φ′(ε)|λ+1(n+1)s(s+1)|φ′(ζ)|λ). $ | (2.5) |
Remark 2.2. If we choose $ \eta(\zeta, \varepsilon) = \zeta-\varepsilon $ with $ \kappa = 1, $ then under the assumption of Theorem 2.2, one has Theorem 2.3 in [19].
Corollary 2.1. If we choose $ \alpha = \kappa = 1, $ and $ n = s = 1, $ then under the assumption of Theoremm 2.2 we have
| $ |16[φ(ε)+4φ(2ε+η(ζ,ε)2)+φ(ε+η(ζ,ε))]−1η(ζ,ε)ε+η(ζ,ε)∫εφ(x)dx|≤η(ζ,ε)4[1+2μ+13μ+1(μ+1)]1μ[(34)1λ+(14)1λ][|φ′(ε)|+|φ′(ζ)|]. $ |
Corollary 2.2. If we choose $ \eta(\zeta, \varepsilon) = \zeta-\varepsilon $ with $ \alpha = \kappa = 1, $ and $ n = s = 1, $ then under the assumption of Theoremm 2.2 we have
| $ |16[φ(ε)+4φ(ε+ζ2)+φ(ζ)]−1ζ−εζ∫εφ(x)dx|≤ζ−ε4[1+2μ+13μ+1(μ+1)]1μ[(34)1λ+(14)1λ][|φ′(ε)|+|φ′(ζ)|]. $ |
Proof. The proof of the last inequality is obtained by using the fact that
| $ n∑i=1(ωi+νi)j≤n∑i=1(ωi)j+n∑i=1(νi)j $ |
for $ 0\leq j\leq1, \, \, \omega_{1}, \omega_{2}, \omega_{3}, ..., \omega_{n}\geq0; \, \, \nu_{1}, \nu_{2}, \nu_{3}, ..., \nu_{n}\geq0. $
Theorem 2.3. Let $ \varphi :\Omega = [\varepsilon, \varepsilon +\eta (\zeta, \varepsilon)]\rightarrow \mathbb{R} $ be a differentiable function on $ \Omega. $ If $ \varphi ^{\prime }\in L[\varepsilon, \varepsilon +\eta (\zeta, \varepsilon)] $and $ |\varphi ^{\prime }| $ is $ s $-preinvex function, then the following inequality holds for fractional integrals with $ \alpha, \kappa > 0, $ then the following integral inequality holds:
| $ |Ψ(ε,ζ;n,α,κ)|≤η(ζ,ε)2(n+1){(Δ1+Δ3)|φ′(ε)|+(Δ1+Δ2)|φ′(ζ)|}, $ | (2.6) |
where
| $ Δ1=2−4(1−2κα2κα+1)ακ+s+13(n+1)s(ακ+s+1)+13(n+1)s[β(ακ+1,s+1)−2β(2κα2κα+1;ακ+1,s+1)], $ |
| $ Δ2=21−s{3−2(2κα2ακ+1)ακ+1−4(1−2κα2ακ+1)ακ+1}3(ακ+1)−2−4(1−2κα2κα+1)ακ+s+13(n+1)s(ακ+s+1)+13(n+1)s[2β(2κα2ακ+1;ακ+1,s+1)−β(ακ+1,s+1)] $ |
and
| $ Δ3={3−2(2κα2ακ+1)ακ+1−4(1−2κα2ακ+1)ακ+1}3(ακ+1)−2−4(1−2κα2κα+1)ακ+s+13(n+1)s(ακ+s+1)+13(n+1)s[2β(2κα2ακ+1;ακ+1,s+1)−β(ακ+1,s+1)]. $ |
Proof. From Lemma 2.1 and using the $ s $-preinvexity of $ |\varphi ^{\prime }(x)|, $ we have
| $ |Ψ(ε,ζ;n,α,κ)|≤η(ζ,ε)2(n+1)[1∫0|2(1−θ)ακ−θακ3|[(1−(1−θn+1)s)|φ′(ε)|+(1−θn+1)s|φ′(ζ)|]dθ+1∫0|θακ−2(1−θ)ακ3|[(1−(n+θn+1)s)|φ′(ε)|+(n+θn+1)s|φ′(ζ)|]dθ] $ | (2.7) |
From (2.7), we have
| $ 1∫0|θακ−2(1−θ)ακ3|[(1−(n+θn+1)s)|φ′(ε)|+(n+θn+1)s|φ′(ζ)|]dθ≤|φ′(ε)|1∫0|θακ−2(1−θ)ακ3|(1−θn+1)sdθ+|φ′(ζ)|1∫0|θακ−2(1−θ)ακ3|(21−s−1−θn+1)sdθ, $ | (2.8) |
By taking into account
| $ Δ1=1∫0|θακ−2(1−θ)ακ3|(1−θn+1)sdθ=2−4(1−2κα2κα+1)ακ+s+13(n+1)s(ακ+s+1)+13(n+1)s[β(ακ+1,s+1)−2β(2κα2κα+1;ακ+1,s+1)], $ | (2.9) |
| $ Δ2=1∫0|θακ−2(1−θ)ακ3|(21−s−1−θn+1)sdθ=21−s{3−2(2κα2ακ+1)ακ+1−4(1−2κα2ακ+1)ακ+1}3(ακ+1)−2−4(1−2κα2κα+1)ακ+s+13(n+1)s(ακ+s+1)+13(n+1)s[2β(2κα2ακ+1;ακ+1,s+1)−β(ακ+1,s+1)] $ | (2.10) |
and
| $ Δ3=1∫0|θακ−2(1−θ)ακ3|(1−1−θn+1)sdθ={3−2(2κα2ακ+1)ακ+1−4(1−2κα2ακ+1)ακ+1}3(ακ+1)−2−4(1−2κα2κα+1)ακ+s+13(n+1)s(ακ+s+1)+13(n+1)s[2β(2κα2ακ+1;ακ+1,s+1)−β(ακ+1,s+1)], $ | (2.11) |
Theorem 2.4. Let $ \varphi :\Omega = [\varepsilon, \varepsilon +\eta (\zeta, \varepsilon)]\rightarrow \mathbb{R} $ be a differentiable function on $ \Omega. $ If $ \varphi ^{\prime }\in L[\varepsilon, \varepsilon +\eta (\zeta, \varepsilon)] $and $ |\varphi ^{\prime }| $ is $ s $-preinvex function, then the following inequality holds for fractional integrals with $ \alpha, \kappa > 0, $ then the following integral inequality holds:
| $ |Ψ(ε,ζ;n,α,κ)|≤η(ζ,ε)2(n+1)(Δ0)1−1λ{(Δ1|φ′(ε)|λ+Δ3|φ′(ζ)|λ)1λ+(Δ1|φ′(ε)|λ+Δ2|φ′(ζ)|λ)1λ}, $ | (2.12) |
where
| $ Δ0={3−2(2κα2ακ+1)ακ+1−4(1−2κα2ακ+1)ακ+1}3(ακ+1). $ |
Proof. By using Lemma 2.1 and the H$ \ddot{o} $lder's integral inequality for $ \lambda > 1 $ we have
| $ |Ψ(ε,ζ;n,α,κ)|≤η(ζ,ε)2(n+1)[(1∫0|2(1−θ)ακ−θακ3|dθ)1−1λ×[1∫0|2(1−θ)ακ−θακ3||φ′(ε+1−θn+1η(ζ,ε))|λdθ)1λ+1∫0|θακ−2(1−θ)ακ3||φ′(ε+n+θn+1η(ζ,ε))|λdθ)1λ] $ |
Using $ s $-preinvexity of $ |\varphi ^{\prime }|, $ we get
| $ |Ψ(ε,ζ;n,α,κ)|≤η(ζ,ε)2(n+1)[(1∫0|2(1−θ)ακ−θακ3|dθ)1−1λ×[1∫0|2(1−θ)ακ−θακ3|[((1−(1−θn+1)s)|φ′(ε)|λ+(1−θn+1)λ|φ′(ζ)|λ)dθ]1λ+1∫0|θακ−2(1−θ)ακ3|[((1−(n+θn+1)s)|φ′(ε)|λ+(n+θn+1)λ|φ′(ζ)|λ)dθ]1λ,] $ | (2.13) |
using the fact that
| $ Δ0=1∫0|2(1−θ)ακ−θακ3|dθ={3−2(2κα2ακ+1)ακ+1−4(1−2κα2ακ+1)ακ+1}3(ακ+1). $ | (2.14) |
A combination of (2.9)–(2.11) with (2.14) into (2.13) gives the desired inequality (2.12). The proof is completed.
Corollary 2.3. If we take $ n = s = 1 $ and $ \alpha = \kappa = 1, $ then under the assumption of Theorem 2.4, we have
| $ |16(φ(ε)+4φ(2ε+η(ζ,ε)2)+φ(ε+η(ζ,ε)))−1η(ζ,ε)ε+η(ζ,ε)∫εφ(x)dx|≤η(ζ,ε)4(518)1−1λ[(61324)1λ+(29324)1λ][|φ′(ε)|+|φ′(ζ)|]. $ |
Corollary 2.4. If we take $ \eta(\zeta, \varepsilon) = \zeta-\varepsilon $ with $ n = s = 1 $ and $ \alpha = \kappa = 1, $ then under the assumption of Theorem 2.4, we have
| $ |16(φ(ε)+4φ(ε+ζ2)+φ(ζ))−1ζ−εζ∫εφ(x)dx|≤ζ−ε4(518)1−1λ[(61324)1λ+(29324)1λ][|φ′(ε)|+|φ′(ζ)|]. $ |
Remark 2.3. If we take $ \eta(\zeta, \varepsilon) = \zeta-\varepsilon $ with $ \lambda = 1 $ and $ \kappa = s = 1, $ then Theorem 2.4 reduces to Theorem 2.2 in [19].
Remark 2.4. If we take $ \eta(\zeta, \varepsilon) = \zeta-\varepsilon $ with $ \lambda = 1, $ $ \kappa = s = 1, $ and $ \alpha = n = 1 $ then Theorem 2.4 reduces to Corollary 1 in [18].
For deriving further results, we deal with the boundedness and the Lipschitzian condition of $ \varphi^{\prime}. $
Theorem 2.5. Let $ \varphi :\Omega = [\varepsilon, \varepsilon +\eta (\zeta, \varepsilon)]\rightarrow \mathbb{R} $ be a differentiable function on $ \Omega. $ If $ |\varphi ^{\prime }| $ is $ s $-preinvex function, there exists constants $ m < \mathcal{M} $ satisfying that $ \infty < m\leq \varphi ^{\prime }(x)\leq \mathcal{M} < \infty $ for all $ x\in \lbrack \varepsilon, \varepsilon +\eta (\zeta, \varepsilon)] $ and $ \alpha, \kappa > 0, $ and $ n\geq 0, $ then the following integral inequality holds:
| $ |Ψ(ε,ζ;n,α,κ)|≤(M−m)η(ζ,ε)2(n+1)Δ0, $ | (2.15) |
where $ \Delta _{0} $ given in (2.14).
Proof. From Lemma 2.1, we have
| $ Ψ(ε,ζ;n,α,κ)|=η(ζ,ε)2(n+1)[1∫0(2(1−θ)ακ−θακ3)(φ′(ε+1−θn+1η(ζ,ε))−m+M2)dθ+1∫0(θακ−2(1−θ)ακ3)(φ′(ε+n+θn+1η(ζ,ε))−m+M2)dθ] $ | (2.16) |
Using the fact that $ m-\frac{m+\mathcal{M}}{2}\leq \varphi ^{\prime }\Big(\varepsilon +\frac{1-\theta }{n+1}\eta (\zeta, \varepsilon)\Big)-\frac{m+ \mathcal{M}}{2}\leq \mathcal{M}-\frac{m+\mathcal{M}}{2}, $ one has
| $ |φ′(ε+1−θn+1η(ζ,ε))−m+M2|≤M−m2, $ |
similarly,
| $ |φ′(ε+n+θn+1η(ζ,ε))−m+M2|≤M−m2, $ |
Inequality (2.16) implies that
| $ |Ψ(ε,ζ;n,α,κ)|≤(M−m)η(ζ,ε)2(n+1)1∫0|2(1−θ)ακ−θακ3|dθ=(M−m)η(ζ,ε)2(n+1)Δ0. $ |
The proof is completed.
Corollary 2.5. If we take $ \eta (\zeta, \varepsilon) = \zeta -\varepsilon $ with $ \alpha = \kappa = 1, $ then under the assumption of Theorem 2.5, we have
| $ |Ψ(ε,ζ;n,α,κ)|≤7(M−m)(ζ−ε)36(n+1). $ |
Theorem 2.6. Let $ \varphi :\Omega = [\varepsilon, \varepsilon +\eta (\zeta, \varepsilon)]\rightarrow \mathbb{R} $ be a differentiable function on $ \Omega. $ If $ \varphi ^{\prime } $ satisfies Lipchitz conditions on $ [\varepsilon, \varepsilon +\eta (\zeta, \varepsilon)] $ for certain $ \mathfrak{L} > 0 $, with $ \alpha > 0, \kappa > 0, \, \, n\geq 0, $ then the following inequality holds:
| $ |Ψ(ε,ζ;n,α,κ)|≤L(η(ζ,ε))22(n+1)2[(n−1)Δ0−4(2ακ2ακ+1)ακ+2−23(ακ)+2+83β(2ακ2ακ+1;ακ+1,2)−43β(ακ+1,2)], $ | (2.17) |
where $ \Delta _{0} $ given in (2.14).
Proof. From Lemma 2.1, we have
| $ Ψ(ε,ζ;n,α,κ)|=η(ζ,ε)2(n+1)[1∫0(2(1−θ)ακ−θακ3)[φ′(ε+1−θn+1η(ζ,ε))−φ′(ε+n+θn+1η(ζ,ε))]dθ. $ |
Since $ \varphi^{\prime} $ is Lipschitz condition on $ [\varepsilon, \varepsilon+\eta(\zeta, \varepsilon)], $ for certain $ \mathfrak{L} > 0, $ we have
| $ |φ′(ε+1−θn+1η(ζ,ε))−φ′(ε+n+θn+1η(ζ,ε))|≤Lη(ζ,ε)(n−1+2tn+1) $ |
Thus
| $ |Ψ(ε,ζ;n,α,κ)|≤η(ζ,ε)2(n+1)[1∫0|2(1−θ)ακ−θακ3||φ′(ε+1−θn+1η(ζ,ε))−φ′(ε+n+θn+1η(ζ,ε)|]dθ≤L(η(ζ,ε))22(n+1)1∫0|2(1−θ)ακ−θακ3|(n−1+2tn+1)dθ=L(η(ζ,ε))22(n+1)2[(n−1)Δ0−4(2ακ2ακ+1)ακ+2−23(ακ)+2+83β(2ακ2ακ+1;ακ+1,2)−43β(ακ+1,2)]. $ |
The proof is completed.
Corollary 2.6. If we take $ \eta(\zeta, \varepsilon) = \zeta-\varepsilon $ with $ \alpha = \kappa = 1, $ then under the assumption of Theorem 2.6, we have
| $ |Ψ(ε,ζ;n,α,κ)|≤L(ζ−ε)22(n+1)2[7n−1118+22135+83β(23;2,2)]. $ |
Let the set $ \psi $ and the $ \delta $-finite measure $ \varrho $ be given, and let the set of all probability densities on $ \varrho $ to be defined on $ \Lambda: = \{z|z:\psi\rightarrow\mathbb{R}, z(x) > 0, \int_{\psi}z(x)d \varrho(x) = 1\}. $
Let $ \mathcal{F}:(0, \infty)\rightarrow\mathbb{R} $ be given mapping and consider $ \mathcal{D}_{\mathcal{F}}(z, w) $ be defined by
| $ DF(z,w):=∫ψz(x)F(z(x)w(x))dϱ(x),z,w∈Λ. $ | (3.1) |
If $ \mathcal{F} $ is convex, then (3.1) is called as the Csiszar $ \mathcal{F} $-divergence.
Consider the following Hermite-Hadamard divergence
| $ DFHH(z,w):=∫ψw(x)z(x)∫1F(t)dtw(x)z(x)−1dϱ(x),w,z∈Λ, $ | (3.2) |
where $ \mathcal{F} $ is convex on $ (0, \infty) $ with $ \mathcal{F}(1) = 0. $ Note that $ \mathcal{D}^{\mathcal{F}}_{HH}(z, w)\geq0 $ with the equality holds if and only if $ w = z. $
Proposition 3.1. Suppose all assumptions of Corollary 2.2 hold for $ (0, \infty) $ and $ \mathcal{F}(1) = 0, $ if $ w, z\in \Lambda, $ then the following inequality holds:
| $ |16[DF(w,z)+4∫ψz(x)F(z(x)+w(x)2w(x)dϱ(x))]−DFHH(w,z)|≤(1+2λ+1)1λ4(3λ+1(1+λ))1λ[(34)1λ+(14)1λ]×[|F′(1)|∫ψ|z(x)−w(x)|dϱ(x)+∫ψ|z(x)−w(x)||F′z(x)w(x)|dϱ(x)]. $ | (3.3) |
Proof. Let $ \Theta _{1} = \{x\in \psi :z(x) = w(x)\}. $ $ \Theta _{2} = \{x\in \psi :z(x) < w(x)\} $ and $ \Theta _{3} = \{x\in \psi :z(x) > w(x)\} $ and Obviously, if $ x\in \Theta _{1}, $ then equality holds in (3.3).
Now if $ x\in \Theta _{2}, $ then using Corollary 2.2 for $ \varepsilon = \frac{z(x)}{w(x)} $ and $ \zeta = 1, $ multiplying both sides of the obtained results by $ w(x) $ and then integrating over $ \Theta _{2}, $ we obtain
| $ |16[4∫ψ2w(x)F(w(x)+z(x)2w(x))dϱ(x)+∫ψ2w(x)F(z(x)w(x))dϱ(x)]−∫ψ2w(x)z(x)w(x)∫1F(t)dtz(x)w(x)−1dϱ(x)|≤(1+2λ+1)1λ4(3λ+1(1+λ))1λ[(34)1λ+(14)1λ]×[|F′(1)|∫ψ2|w(x)−z(x)|dϱ(x)+∫ψ2|w(x)−z(x)||F′z(x)w(x)|dϱ(x)]. $ | (3.4) |
Similarly if $ x\in \Theta _{3}, $ then using Corollary 2.2 for $ \varepsilon = 1 $ and $ \zeta = \frac{z(x)}{w(x)}, $ multiplying both sides of the obtained results by $ w(x) $ and then integrating over $ \Theta _{3}, $ we obtain
| $ |16[4∫ψ3w(x)F(w(x)+z(x)2w(x))dϱ(x)+∫ψ3w(x)F(z(x)w(x))dϱ(x)]−∫ψ3w(x)z(x)w(x)∫1F(t)dtz(x)w(x)−1dϱ(x)|≤(1+2λ+1)1λ4(3λ+1(1+λ))1λ[(34)1λ+(14)1λ]×[|F′(1)|∫ψ3|z(x)−w(x)|dϱ(x)+∫ψ3|z(x)−w(x)||F′z(x)w(x)|dϱ(x)]. $ | (3.5) |
Adding inequalities (3.4) and (3.5) and then utilizing triangular inequality, we get the result.
Proposition 3.2. Suppose all assumptions of Corollary 2.4 holds for $ (0, \infty) $ and $ f(1) = 0, $ if $ w, z\in \Lambda, $ then the following inequality holds:
| $ |16[DF(w,z)+4∫ψz(x)F(z(x)+w(x)2w(x)dϱ(x))]−DFHH(w,z)|≤(518)1λ[(61324)1λ+(29324)1λ]×[|F′(1)|∫ψ|z(x)−w(x)|4dϱ(x)+∫ψ|z(x)−w(x)|4|F′z(x)w(x)|dϱ(x)]. $ | (3.6) |
Proof. The proof is similar as one has done for Corollary 2.2.
Let $ \mathfrak{G}:[\varepsilon, \varepsilon+\eta(\zeta, \varepsilon)] \rightarrow[0, 1] $ be the probability density function of a continuous random variable $ Y $ with the cumulative distribution function of $ \mathfrak{G} $
| $ F(y)=P(Y≤y)=x∫εG(t)dt. $ |
As we know $ E(y) = \int\limits_{\varepsilon}^{\varepsilon+\eta(\zeta, \varepsilon)}tdF(t) = \big(\varepsilon+\eta(\zeta, \varepsilon)\big)- \int\limits_{\varepsilon}^{\varepsilon+\eta(\zeta, \varepsilon)}F(t) $
Proposition 3.3. By Corollary 2.1, we get the inequality
| $ |16[4P(Y≤2ε+η(ζ,ε)2)+1]−1η(ζ,ε)(ε+η(ζ,ε)−E(y))|≤η(ζ,ε)4[1+2μ+13μ+1(μ+1)]1μ[(34)1λ+(14)1λ][|G(ε)|+|G(ζ)|]. $ |
Proposition 3.4. By Corollary 2.3, we get the inequality
| $ |16[4P(Y≤2ε+η(ζ,ε)2)+1]−1η(ζ,ε)(ε+η(ζ,ε)−E(y))|≤η(ζ,ε)4(518)1−1λ[(61324)1λ+(29324)1λ][|G(ε)|+|G(ζ)|]. $ |
Same applications can be found for Corollary 2.2 and Corollary 2.4, respectively. We leave it to the interested readers.
In this paper, we have established several Simpson's type inequalities via $ \kappa $-fractional integrals in terms of preinvex functions. We also have obtained the inequalities applied to $ \mathcal{F} $-divergence measures and application for probability density functions. These results can be viewed as refinement and significant improvements of the previously known for [18,19] and preinvex functions. Applications can be provided in terms of the obtained results to special means. The ideas and techniques of this paper may be attracted to interested readers.
The authors declare that there is no conflicts of interest in this paper.
| [1] |
Banat IM, Franzetti A, Gandolfi I, et al. (2010) Microbial biosurfactants production, applications and future potential. Appl Microbiol Biotechnol 87: 427-444. doi: 10.1007/s00253-010-2589-0
|
| [2] |
Chen ML, Penfold J, Thomas R.K, et al. (2010) Mixing behaviour of the biosurfactant, rhamnolipid, with a conventional anionic surfactant, sodium dodecyl benzene sulfonate. Langmuir 26: 17958-17968. doi: 10.1021/la1031834
|
| [3] |
Chen ML, Penfold J, Thomas RK, et al. (2010) Solution self-assembly and adsorption at the air-water interface of the monorhamnose and dirhamnose rhamnolipids and their mixtures. Langmuir 26: 18281-18292. doi: 10.1021/la1031812
|
| [4] | Fracchia L, Ceresa C, Franzetti A, et al. (2014) Industrial applications of biosurfactants, In: N. Kosaric, F.V. Sukan (Ed), Biosurfactant—Production and Utilization—Processes, Technologies, and Economics, Boca Raton: CRS Press—Taylor & Francis Group, 245-267. |
| [5] |
Banat IM, Makkar RS, Cameotra SS (2000) Potential commercial applications of microbial surfactants. Appl Microbiol Biotechnol 53: 495-508. doi: 10.1007/s002530051648
|
| [6] | Fracchia L, Cavallo M, Martinotti MG, et al. (2012) Biosurfactants and bioemulsifiers, biomedical and related applications-present status and future potentials, In: D.N. Ghista (Ed), Biomedical Science, Engineering and Technology, Rijeka: InTech, 325-370. |
| [7] |
Ortiz A, Teruel JA, Espuny MJ, et al. (2009) Interactions of a bacterial biosurfactant trehalose lipid with phosphatidylserine membranes. Chem Phys Lipids 158: 46-53. doi: 10.1016/j.chemphyslip.2008.11.001
|
| [8] |
Sánchez M, Aranda FJ, Teruel JA, et al. (2010) Permeabilization of biological and artificial membranes by a bacterial dirhamnolipid produced by Pseudomonas aeruginosa. J Colloid Interface Sci 341: 240-247. doi: 10.1016/j.jcis.2009.09.042
|
| [9] |
Sotirova AV, Spasova DI, Galabova DN, et al. (2008) Rhamnolipid biosurfactant permeabilizing effects on gram-positive and gram-negative bacterial strains. Curr Microbiol 56: 639-644. doi: 10.1007/s00284-008-9139-3
|
| [10] | Zaragoza A, Aranda FJ, Espuny MJ, et al. (2010) Hemolytic activity of a bacterial trehalose lipid biosurfactant produced by Rhodococcus sp., evidence for a colloid-osmotic mechanism. Langmuir 26: 8567-8572. |
| [11] |
Banat IM, Rienzo MAD, Quinn GA (2014) Microbial biofilms, biosurfactants as antibiofilm agents. Appl Microbiol Biotechnol 98: 9915-9929. doi: 10.1007/s00253-014-6169-6
|
| [12] |
Cochis A, Fracchia L, Martinotti MG, et al. (2012) Biosurfactants prevent in‐vitro C. albicans biofilm formation on resins and silicon materials for prosthetic devices. Oral Surg Oral Med Oral Pathol Oral Radiol 113: 755-761. doi: 10.1016/j.oooo.2011.11.004
|
| [13] | Muthusamy K, Gopalakrishnan S, Ravi TK, et al. (2008) Biosurfactants, properties, commercial production and application. Curr Sci 94: 736-747. |
| [14] |
Bĕhal V (2006) Mode of action of microbial bioactive metabolites. Folia Microbiol 51: 359-369. doi: 10.1007/BF02931577
|
| [15] |
Quinn GA, Maloy AP, Banat MM, et al. (2013) A comparison of effects of broad-spectrum antibiotics and biosurfactants on established bacterial biofilms. Curr Microbiol 67: 614-623. doi: 10.1007/s00284-013-0412-8
|
| [16] |
Rodrigues L, Banat IM, Teixeira J, et al. (2007) Strategies for the prevention of microbial biofilm formation on silicone rubber voice prostheses. J Biomed Mater Res B Appl Biomater 81B: 358-370. doi: 10.1002/jbm.b.30673
|
| [17] |
Carrillo C, Teruel JA, Aranda FA, et al. (2003) Molecular mechanism of membrane permeabilization by the peptide antibiotic surfactin. Biochem Biophys Acta 1611: 91-97. doi: 10.1016/S0005-2736(03)00029-4
|
| [18] |
Deleu M, Paquot M, Nylander T (2008) Effect of fengycin, a lipopeptide produced by Bacillus subtilis on model biomembranes. Biophys J 94: 2667-2679. doi: 10.1529/biophysj.107.114090
|
| [19] |
Horn JN, Sengillo JD, Lin D, et al. (2012) Characterization of a potent antimicrobial lipopeptide via coarse-grained molecular dynamics. Biochim Biophys Acta 1818: 212-218. doi: 10.1016/j.bbamem.2011.07.025
|
| [20] |
Mandal SM, Barbosa AE, Franco OL (2013) Lipopeptides in microbial infection control, scope and reality for industry. Biotechnol Adv 31: 338-345. doi: 10.1016/j.biotechadv.2013.01.004
|
| [21] |
Scott WR, Baek SB, Jung D, et al. (2007) NMR structural studies of the antibiotic lipopeptide daptomycin in DHPC micelles. Biochim Biophys Acta 1768: 3116-3126. doi: 10.1016/j.bbamem.2007.08.034
|
| [22] |
Mangoni ML, Shai Y (2011) Short native antimicrobial peptides and engineered ultrashort lipopeptides, similarities and differences in cell specificities and modes of action. Cell Mol Life Sci 68: 2267-2280. doi: 10.1007/s00018-011-0718-2
|
| [23] |
Zaragoza A, Aranda FJ, Espuny MJ, et al. (2009) A mechanism of membrane permeabilization by a bacterial trehalose lipid biosurfactant produced by Rhodococcus sp. Langmuir 25: 7892-7898. doi: 10.1021/la900480q
|
| [24] | Cochrane SA, Vederas JC (2014) Lipopeptides from Bacillus and Paenibacillus spp.: A Gold Mine of Antibiotic Candidates. Med Res Rev DOI 10.1002/med.21321. |
| [25] |
Soon RL, Velkov T, Chiu F, et al. (2011) Design, synthesis, and evaluation of a new fluorescent probe for measuring polymyxin lipopolysaccharide binding interactions. Anal Biochem 409: 273-283. doi: 10.1016/j.ab.2010.10.033
|
| [26] |
Velkov T, Thompson PE, Nation RL, et al. (2010) Structure-activity relationships of polymyxin antibiotics. J Med Chem 53: 1898-1916. doi: 10.1021/jm900999h
|
| [27] |
Maget-Dana R, Harnois I, Ptak M (1989) Interactions of the lipopeptide antifungal iturin A with lipids in mixed monolayers. Biochim Biophys Acta 981: 309-314. doi: 10.1016/0005-2736(89)90042-4
|
| [28] |
Sotirova A, Spasova D, Vasileva-Tonkova E, et al. (2009) Effects of rhamnolipid-biosurfactant on cell surface of Pseudomonas aeruginosa. Microbiol Res 164: 297-303. doi: 10.1016/j.micres.2007.01.005
|
| [29] | Seung-Hak B, Sun XX, Lee YJ, et al. (2003) Mitigation of harmful algae blooms by sophorolipid. J Microbiol Biotechnol 13: 651-659. |
| [30] | Rodrigues L, van der Mei HC, Banat IM, et al. (2006) Inhibition of microbial adhesion to silicone rubber treated with biosurfactant from Streptococcus thermophilus A. FEMS Immunol Med Microbiol 46: 107-112. |
| [31] |
Rodrigues LR, Banat IM, van der Mei HC, et al. (2006) Interference in adhesion of bacteria and yeasts isolated from explanted voice prostheses to silicone rubber by rhamnolipid biosurfactants. J Appl Microbiol 100: 470-480. doi: 10.1111/j.1365-2672.2005.02826.x
|
| [32] |
Vater J, Kablitz B, Wilde C, et al. (2002) Matrix-assisted laser desorption ionization time of flight mass spectrometry of lipopeptide biosurin whole cells and culture filtrates of Bacillus subtilis C-1 isolated from petroleum sludge. Appl Environ Microbiol 68: 6210-6219. doi: 10.1128/AEM.68.12.6210-6219.2002
|
| [33] |
Baltz RH, Miao V, Wrigley SK (2005) Natural products to drugs, daptomycin and related lipopeptide antibiotics. Nat Prod Rep 22: 717-741. doi: 10.1039/b416648p
|
| [34] |
Landman D, Georgescu C, Martin DA, et al. (2008) Polymyxins revisited. Clin Microbiol Rev 21: 449-465. doi: 10.1128/CMR.00006-08
|
| [35] |
Saini HS, Barragán-Huerta BE, Lebrón-Paler A, et al. (2008) Efficient purification of the biosurfactant viscosin from Pseudomonas libanensis strain M9-3 and its physicochemical and biological properties. J Nat Prod 71: 1011-1015. doi: 10.1021/np800069u
|
| [36] |
Benincasa M, Abalos A, Oliveira I, et al. (2004) Chemical structure, surface properties and biological activities of the biosurfactant produced by Pseudomonas aeruginosa LBI from soapstock. Antonie Van Leeuwenhoek 85: 1-8. doi: 10.1023/B:ANTO.0000020148.45523.41
|
| [37] | De Rienzo MAD, Banat IM, Dolman B, et al. (2015) Sophorolipid biosurfactants, antibacterial activities and characteristics. New Biotechnol DOI: 10.1016/j.nbt.2015.02.009. |
| [38] | Kim K, Yoo D, Kim Y, et al. (2002) Characteristics sophorolipid as an antimicrobial agent. J Microbiol Biotechnol 12: 235-241. |
| [39] |
Kitamoto D, Yanagishita H, Shinbo T, et al. (1993) Surface active properties and antimicrobial activities of mannosylerythritol lipids as biosurfactants produced by Candida antarctica. J Biotechnol 29: 91-96. doi: 10.1016/0168-1656(93)90042-L
|
| [40] | Ghribi D, Abdelkefi-Mesrati L, Mnif I, et al. (2012) Investigation of antimicrobial activity and statistical optimization of Bacillus subtilis SPB1 biosurfactant production in solid-state fermentation. J Biomed Biotechnol DOI: 10.1155/2012/373682. |
| [41] |
Ding R, Wu XC, Qian CD, et al. (2011) Isolation and identification of lipopeptide antibiotics from Paenibacillus elgii B69 with inhibitory activity against methicillin-resistant Staphylococcus aureus. J Microbiol 49: 942-949. doi: 10.1007/s12275-011-1153-7
|
| [42] |
Tabbene O, Kalai L, Ben Slimene I, et al. (2011) Anti-candida effect of bacillomycin D-like lipopeptides from Bacillus subtilis B38. FEMS Microbiol Lett 316: 108-114. doi: 10.1111/j.1574-6968.2010.02199.x
|
| [43] |
Gomaa EZ (2013) Antimicrobial activity of a biosurfactant produced by Bacillus licheniformis strain M104 grown on whey. Braz Arch Biol Technol 56: 259-268. doi: 10.1590/S1516-89132013000200011
|
| [44] |
Song B, Rong Y-J, Zhao M-X, et al. (2013) Antifungal activity of the lipopeptides produced by Bacillus amyloliquefaciens anti-CA against Candida albicans isolated from clinic. Appl Microbiol Biotechnol 97: 7141-7150. doi: 10.1007/s00253-013-5000-0
|
| [45] | Sharma D, Mandal SM, Manhas RK (2014) Purification and characterization of a novel lipopeptide from Streptomyces amritsarensis sp. nov. active against methicillin-resistant Staphylococcus aureus. AMB Express 4: 50-58. |
| [46] |
Liang TW, Wu CC, Cheng WT, et al. (2014) Exopolysaccharides and antimicrobial biosurfactants produced by Paenibacillus macerans TKU029. Appl Biochem Biotechnol 172: 933-950. doi: 10.1007/s12010-013-0568-5
|
| [47] |
Wasserman HH, Keggi JJ, Mckeon JE (1962) The structure of serratamolide. J Am Chem Soc 84: 2978-2982. doi: 10.1021/ja00874a028
|
| [48] | Escobar-Diaz E, Lopez-Martin EM, Hernandez del Cerro M, et al. (2005) AT514, a cyclic depsipeptide from Serratia marcescens, induces apoptosis of B-chronic lymphocytic leukemia cells: interference with the Akt/NF-kappaB survival pathway. Leukemia 19: 572-579. |
| [49] | Tomas RP, Ramoneda BM, Lledo EG, et al. (2005) Use of cyclic depsipeptide as a chemotherapeutic agent against cancer. Patent Number: EP1553080. |
| [50] | Strobel GA, Morrison SL, Cassella M (2005) Protecting plants from oomycete pathogens by treatment with compositions containing serratamolide and oocydin a from Serratia marcescens. Patent Number: US2003049230-A1; US6926892-B2. |
| [51] |
Kadouri DE, Shanks RM (2013) Identification of a methicillin-resistant Shaphylococcus aureus inhibitory compound isolated from Serratia marcescens. Res Microbiol 164: 821-826. doi: 10.1016/j.resmic.2013.06.002
|
| [52] |
Samadi N, Abadian N, Ahmadkhaniha R, et al. (2012) Structural characterization and surface activities of biogenic rhamnolipid surfactants from Pseudomonas aeruginosa isolate MN1 and synergistic effects against methicillin-resistant Staphylococcus aureus. Folia Microbiol 57: 501-508. doi: 10.1007/s12223-012-0164-z
|
| [53] |
Magalhães L, Nitschke M (2013) Antimicrobial activity of rhamnolipids against Listeria monocytogenes and their synergistic interaction with nisin. Food Control 29: 138-142. doi: 10.1016/j.foodcont.2012.06.009
|
| [54] | Luna JM, Rufino RD, Campos-Takaki GM, et al. (2012) Properties of the biosurfactant produced by Candida sphaerica cultivated in low-cost substrates. Chem Eng Trans 27: 67-72. |
| [55] |
Rufino RD, Luna JM, Sarubbo LA, et al. (2011) Antimicrobial and anti-adhesive potential of a biosurfactant Rufisan produced by Candida lipolytica UCP 0988. Colloids Surf B 84: 1-5. doi: 10.1016/j.colsurfb.2010.10.045
|
| [56] | Joshi-Navare K, Prabhune A. (2013) A Biosurfactant-Sophorolipid Acts in Synergy with Antibiotics to Enhance Their Efficiency. BioMed Research Int DOI: 10.1155/2013/512495. |
| [57] | Donio MBS, Ronica FA, Viji VT, et al. (2013) Halomonas sp. BS4, A biosurfactant producing halophilic bacterium isolated from solar salt works in India and their biomedical importance. Springer Plus 2: 149-159. |
| [58] |
Ngai AL, Bourque MR, Lupinacci RJ, et al. (2011) Overview of safety experience with caspofungin in clinical trials conducted over the first 15 years, A brief report. Int J Antimicrob Ag 38: 540-544. doi: 10.1016/j.ijantimicag.2011.07.008
|
| [59] |
Emiroglu M (2011) Micafungin use in children. Expert Rev Anti-Infect Ther 9: 821-834. doi: 10.1586/eri.11.91
|
| [60] | George J, Reboli AC (2012) Anidulafungin, When and how? The clinician's view. Mycoses 55: 36-44. |
| [61] |
Robbel L, Marahiel MA (2010) Daptomycin, a bacterial lipopeptide synthesized by a nonribosomal machinery. J Biol Chem 285: 27501-27508. doi: 10.1074/jbc.R110.128181
|
| [62] |
Tally FP, Zeckel M, Wasilewski MM, et al. (1999) Daptomycin, A novel agent for Gram-positive infections. Expert Opin Inv Drug 8: 1223-1238. doi: 10.1517/13543784.8.8.1223
|
| [63] |
Wagner C, Graninger W, Presterl E, et al. (2006) The echinocandins, Comparison of their pharmacokinetics, pharmacodynamics and clinical applications. Pharmacology 78: 161-177. doi: 10.1159/000096348
|
| [64] |
Denning DW (2002) Echinocandins, A new class of antifungal. J Antimicrob Chemoth 49: 889-891. doi: 10.1093/jac/dkf045
|
| [65] | Hill J, Parr I, Morytko M, et al. (2008) Lipopeptides as antibacterial agents. US Patent US7335725 B2, February 26. |
| [66] | Burke T, Chandrasekhar B, Knight M (1999) Analogues of viscosin and uses thereof. United States Patent US5965524, October 12. |
| [67] | Seydlová G, Svobodová J (2008) Review of surfactin chemical properties and the potential biomedical applications. Cent Eur J Med 3: 123-133. |
| [68] |
Huang X, Lu Z, Zhao H, et al. (2006) Antiviral activity of antimicrobial lipopeptide from Bacillus subtilis fmbj against pseudorabies virus, porcine parvovirus, newcastle disease virus and infectious bursal disease virus in vitro. Int J Pept Res Ther 12: 373-377. doi: 10.1007/s10989-006-9041-4
|
| [69] |
Shah V, Doncel GF, Seyoum T, et al. (2005) Sophorolipids, microbial glycolipids with anti-human immunodeficiency virus and sperm-immobilizing activities. Antimicrob Agents Chemother 49: 4093-4100. doi: 10.1128/AAC.49.10.4093-4100.2005
|
| [70] | Remichkova M, Galabova D, Roeva I, et al. (2008) Anti-herpesvirus activities of Pseudomonas sp. S-17 rhamnolipid and its complex with alginate. Z Naturforsch C 63: 75-81. |
| [71] |
Donlan RM, Costerton JW (2002) Biofilms, survival mechanisms of clinically relevant microorganisms. Clin Microbiol Rev 15: 167-193. doi: 10.1128/CMR.15.2.167-193.2002
|
| [72] |
Kurtz S, Ong K, Lau E, et al. (2007) Projections of primary and revision hip and knee arthroplasty in the United States from 2005 to 2030. J Bone Joint Surg Am 89: 780-785. doi: 10.2106/JBJS.F.00222
|
| [73] | Hamilton H, Jamieson J (2008) Deep infection in total hip arthroplasty. Can J Surg 51: 111-117. |
| [74] | Francolini I, Donelli G (2010) Prevention and control of biofilm-based medical-device-related infections, FEMS Immunol Med Microbiol 59: 227-238. |
| [75] |
Pinto S, Alves P, Matos CM, et al. (2010) Poly(dimethyl siloxane) surface modification by low pressure plasma to improve its characteristics towards biomedical applications. Colloids Surf B 81: 20-26. doi: 10.1016/j.colsurfb.2010.06.014
|
| [76] |
Makamba H, Kim JH, Lim K, et al. (2003) Surface modification of poly(dimethyl siloxane microchannels. Electrophoresis 24: 3607-3619. doi: 10.1002/elps.200305627
|
| [77] |
Vasilev K, Cook J, Griesser HJ (2009) Antibacterial surfaces for biomedical devices. Expert. Rev Med Devices 6: 553-567. doi: 10.1586/erd.09.36
|
| [78] |
de Sainte Claire P (2009) Degradation of PEO in the Solid State, A Theoretical Kinetic Model. Macromolecules 42: 3469-3482. doi: 10.1021/ma802469u
|
| [79] |
Hegstad K, Langsrud S, Lunestad BT, et al. (2010) Does the wide use of quaternary ammonium compounds enhance the selection and spread of antimicrobial resistance and thus threaten our health? Microb Drug Resist 16: 91-104. doi: 10.1089/mdr.2009.0120
|
| [80] | Kiran GS, Sabarathnam B, Selvin J (2010) Biofilm disruption potential of a glycolipid biosurfactant from marine Brevibacterium casei. FEMS Immunol Med Microbiol 59: 432-8. |
| [81] |
Rivardo F, Martinotti MG, Turner RJ, et al. (2011) Synergistic effect of lipopeptide biosurfactant with antibiotics against Escherichia coli CFT073 biofilm. Int J Antimicrob Ag 37: 324-331. doi: 10.1016/j.ijantimicag.2010.12.011
|
| [82] | Ceri H, Turner R, Martinotti MG, et al. (2010) Biosurfactant composition produced by a new Bacillus licheniformis strain, uses and products thereof. World Patent WO2010067345A1. |
| [83] | Janek T, Łukaszewicz M, Krasowska A (2012) Antiadhesive activity of the biosurfactant pseudofactin II secreted by the Arctic bacterium Pseudomonas fluorescens BD5. BMC Microbiology DOI: 10.1186/1471-2180-12-24 |
| [84] | Quinn GA, Maloy AP, McClean S, et al. (2012) Lipopeptide biosurfactants from Paenibacillus polymyxa inhibits single and mixed species biofilms. Biofouling 8: 1151-1156. |
| [85] |
Sriram MI, Kalishwaralal K, Deepak V, et al. (2011) Biofilm inhibition and antimicrobial action of lipopeptide biosurfactant produced by heavy metal tolerant strain Bacillus cereus NK1. Colloids Surf B 85: 174-181. doi: 10.1016/j.colsurfb.2011.02.026
|
| [86] |
Zeraik AE, Nitschke M (2010) Biosurfactants as agents to reduce adhesion of pathogenic bacteria to polystyrene surfaces, effect of temperature and hydrophobicity. Curr Microbiol 61: 554-559. doi: 10.1007/s00284-010-9652-z
|
| [87] |
Pradhan AK, Pradhan N, Mall G, et al. (2013) Application of lipopeptide biosurfactant isolated from a halophile, Bacillus tequilensis CH for inhibition of biofilm. Appl Biochem Biotechnol 171: 1362-1375 doi: 10.1007/s12010-013-0428-3
|
| [88] | Ceresa C, Tessarolo F, Caola I, et al. (2015) Inhibition of Candida albicans adhesion on medical-grade silicone by a Lactobacillus-derived biosurfactant. J Appl Microbiol 18:1116-1125. |
| [89] |
Singh N, Pemmaraju SC, Pruthi PA, et al. (2013) Candida biofilm disrupting ability of di-rhamnolipid (RL-2) produced from Pseudomonas aeruginosa DSVP20. Appl Biochem Biotechnol 169: 2374-2391. doi: 10.1007/s12010-013-0149-7
|
| [90] |
Pradhan AK, Pradhan N, Sukla LB, et al. (2014) Inhibition of pathogenic bacterial biofilm by biosurfactant produced by Lysinibacillus fusiformis S9. Bioprocess Biosyst Eng 37: 139-149. doi: 10.1007/s00449-013-0976-5
|
| [91] |
Nickzad A, Deziel E (2014) The involvement of rhamnolipids in microbial cell adhesion and biofilm development- an approach for control? Lett Appl Microbiol 58: 447-453. doi: 10.1111/lam.12211
|
| [92] | Padmapriya B, Suganthi S (2013) Antimicrobial and anti adhesive activity of purified biosurfactants produced by Candida species. Middle-East J Sci Res 14: 1359-1369. |
| [93] |
Tahmourespour A, Salehi R, Kermanshahi RK (2011) Lactobacillus acidophilus-derived biosurfactant effect on gtfB and gtfC expression level in Streptococcus mutans biofilm cells. Braz J Microbiol 42: 330-339. doi: 10.1590/S1517-83822011000100042
|
| [94] | Bruce AW, Busscher HJ, Reid G, et al. (2000) Lactobacillus therapies, U.S. Patent US6051552A. |
| [95] |
Hajfarajollah H, Mokhtarani B, Noghabi KA (2014) Newly antibacterial and antiadhesive lipopeptide biosurfactant secreted by a probiotic strain, Propionibacterium freudenreichii. Appl Biochem Biotechnol 174: 2725-2740. doi: 10.1007/s12010-014-1221-7
|
| [96] |
Cao XH, Wang AH, Wang CL, et al. (2010) Surfactin induces apoptosis in human breast cancer MCF-7 cells through a ROS/JNK-mediated mitochondrial/caspase pathway. Chem Biol Interact 183: 357-362. doi: 10.1016/j.cbi.2009.11.027
|
| [97] | Escobar-Díaz E, López-Martín EM, Hernández del Cerro M, et al. (2005) AT514, a cyclic depsipeptide from Serratia marcescens, induces apoptosis of B-chronic lymphocytic leukemia cells, interference with the Akt/NF-kappaB survival pathway. Leukemia 19: 572-579. |
| [98] |
Kitamoto D, Isoda H, Nakahara T (2002) Functions and potential applications of glycolipid biosurfactants-from energy-saving materials to gene delivery carriers. J Biosci Bioeng 94: 187-201. doi: 10.1016/S1389-1723(02)80149-9
|
| [99] |
Chen J, Song X, Zhang H, et al. (2006) Sophorolipid produced from the new yeast strain Wickerhamiella domercqiae induces apoptosis in H7402 human liver cancer cells. Appl Microbiol Biotechnol 72: 52-59 doi: 10.1007/s00253-005-0243-z
|
| [100] |
Tang JS, Zhao F, Gao H, et al. (2010) Characterization and online detection of surfactin isomers based on HPLC-MS analyses and their inhibitory effects on the overproduction of nitric oxide and the release of TNF-α and IL-6 in LPS-induced macrophages. Mar Drugs 8: 2605-2618. doi: 10.3390/md8102605
|
| [101] |
Park SY, Kim YH, Kim EK, et al. (2010) Heme oxygenase-1 signals are involved in preferential inhibition of pro-inflammatory cytokine release by surfactin in cells activated with Porphyromonas gingivalis lipopolysaccharide. Chem Biol Interact 188: 437-45. doi: 10.1016/j.cbi.2010.09.007
|
| [102] |
Park SY, Kim Y (2009) Surfactin inhibits immunostimulatory function of macrophages through blocking NK-κB, MAPK and Akt pathway. Int Immunopharmacol 9: 886-893. doi: 10.1016/j.intimp.2009.03.013
|
| [103] |
Faivre V, Rosilio V (2010) Interest of glycolipids in drug delivery, from physicochemical properties to drug targeting. Expert Opin Drug Del 7: 1031-1048. doi: 10.1517/17425247.2010.511172
|
| [104] |
Nguyen TTL, Edelen A, Neighbors B, et al. (2010) Biocompatible lecithin-based microemulsions with rhamnolipid and sophorolipid biosurfactants, Formulation and potential applications. J Colloid Interf Sci 348: 498-504. doi: 10.1016/j.jcis.2010.04.053
|
| [105] | Nicoli S, Eeman M, Deleu M, et al. (2010) Effect of lipopeptides and iontophoresis on aciclovir skin delivery. J Pharm Pharmacol 62: 702-708. |
| [106] |
Plaza GA, Chojniak J, Banat IM (2014) Biosurfactant mediated biosynthesis of selected metallic nanoparticles. Int J Molecular Sci 15: 13720-13737. doi: 10.3390/ijms150813720
|
| [107] |
Reddy AS, Chen CY, Chen CC, et al. (2010) Biological synthesis of gold and silver nanoparticles mediated by the bacteria Bacillus subtilis. J Nanosci Nanotechnol 10: 6567-6574. doi: 10.1166/jnn.2010.2519
|
| [108] |
Singh BR, Dwivedi S, Al-Khedhairy AA, et al. (2011) Synthesis of stable cadmium sulfide nanoparticles using surfactin produced by Bacillus amyloliquifaciens strain KSU-109. Colloids Surf B 85: 207-213. doi: 10.1016/j.colsurfb.2011.02.030
|
| [109] |
Palanisamy P, Raichur AM (2009) Synthesis of spherical NiO nanoparticles through a novel biosurfactant mediated emulsion technique. Mater Sci Eng C Biomim Supramol Syst 29: 199-204. doi: 10.1016/j.msec.2008.06.008
|
| [110] |
Kumar CG, Mamidyala SK, Das B, et al. (2010) Synthesis of biosurfactant-based silver nanoparticles with purified rhamnolipids isolated from Pseudomonas aeruginosa BS-161R. J Microbiol Biotechnol 20: 1061-1068. doi: 10.4014/jmb.1001.01018
|
| [111] |
D'Britto V, Kapse H, Babrekar H, et al. (2011) Silver nanoparticles studded porous polyethylene scaffolds, bacteria struggle to grow on them while mammalian cells thrive. Nanoscale 3: 2957-2963. doi: 10.1039/c1nr10154d
|
| 1. | Saima Rashid, Muhammad Aslam Noor, Khalida Inayat Noor, Farhat Safdar, Yu-Ming Chu, Hermite-Hadamard Type Inequalities for the Class of Convex Functions on Time Scale, 2019, 7, 2227-7390, 956, 10.3390/math7100956 | |
| 2. | Saima Rashid, Fahd Jarad, Muhammad Aslam Noor, Humaira Kalsoom, Yu-Ming Chu, Inequalities by Means of Generalized Proportional Fractional Integral Operators with Respect to Another Function, 2019, 7, 2227-7390, 1225, 10.3390/math7121225 | |
| 3. | Saima Rashid, Fahd Jarad, Muhammad Aslam Noor, Grüss-type integrals inequalities via generalized proportional fractional operators, 2020, 114, 1578-7303, 10.1007/s13398-020-00823-5 | |
| 4. | Muhammad Samraiz, Fakhra Nawaz, Sajid Iqbal, Thabet Abdeljawad, Gauhar Rahman, Kottakkaran Sooppy Nisar, Certain mean-type fractional integral inequalities via different convexities with applications, 2020, 2020, 1029-242X, 10.1186/s13660-020-02474-x | |
| 5. | Thabet Abdeljawad, Saima Rashid, Zakia Hammouch, İmdat İşcan, Yu-Ming Chu, Some new Simpson-type inequalities for generalized p-convex function on fractal sets with applications, 2020, 2020, 1687-1847, 10.1186/s13662-020-02955-9 | |
| 6. | Zeynep ŞANLI, SIMPSON TYPE INTEGRAL INEQUALITIES FOR HARMONIC CONVEX FUNCTIONS VIA CONFORMABLE FRACTIONAL INTEGRALS, 2020, 2147-1630, 10.37094/adyujsci.780433 | |
| 7. | Thabet Abdeljawad, 2020, 9781119654223, 133, 10.1002/9781119654223.ch5 | |
| 8. | Shuang-Shuang Zhou, Saima Rashid, Saima Parveen, Ahmet Ocak Akdemir, Zakia Hammouch, New computations for extended weighted functionals within the Hilfer generalized proportional fractional integral operators, 2021, 6, 2473-6988, 4507, 10.3934/math.2021267 | |
| 9. | Thabet Abdeljawad, Saima Rashid, Zakia Hammouch, Yu-Ming Chu, Some new local fractional inequalities associated with generalized $(s,m)$-convex functions and applications, 2020, 2020, 1687-1847, 10.1186/s13662-020-02865-w | |
| 10. | Humaira Kalsoom, Saima Rashid, Muhammad Idrees, Yu-Ming Chu, Dumitru Baleanu, Two-Variable Quantum Integral Inequalities of Simpson-Type Based on Higher-Order Generalized Strongly Preinvex and Quasi-Preinvex Functions, 2019, 12, 2073-8994, 51, 10.3390/sym12010051 | |
| 11. | Saima Rashid, Fahd Jarad, Zakia Hammouch, Some new bounds analogous to generalized proportional fractional integral operator with respect to another function, 2021, 0, 1937-1179, 0, 10.3934/dcdss.2021020 | |
| 12. | Saima Rashid, Fahd Jarad, Muhammad Aslam Noor, Khalida Inayat Noor, Dumitru Baleanu, Jia-Bao Liu, On Grüss inequalities within generalized K-fractional integrals, 2020, 2020, 1687-1847, 10.1186/s13662-020-02644-7 | |
| 13. | Saad Ihsan Butt, Mehroz Nadeem, Shahid Qaisar, Ahmet Ocak Akdemir, Thabet Abdeljawad, Hermite–Jensen–Mercer type inequalities for conformable integrals and related results, 2020, 2020, 1687-1847, 10.1186/s13662-020-02968-4 | |
| 14. | Saima Rashid, Muhammad Amer Latif, Zakia Hammouch, Yu-Ming Chu, Fractional Integral Inequalities for Strongly h -Preinvex Functions for a kth Order Differentiable Functions, 2019, 11, 2073-8994, 1448, 10.3390/sym11121448 | |
| 15. | SAIMA RASHID, ZAKIA HAMMOUCH, FAHD JARAD, YU-MING CHU, NEW ESTIMATES OF INTEGRAL INEQUALITIES VIA GENERALIZED PROPORTIONAL FRACTIONAL INTEGRAL OPERATOR WITH RESPECT TO ANOTHER FUNCTION, 2020, 28, 0218-348X, 2040027, 10.1142/S0218348X20400277 | |
| 16. | Fatih Hezenci, Hüseyin Budak, Hasan Kara, New version of fractional Simpson type inequalities for twice differentiable functions, 2021, 2021, 1687-1847, 10.1186/s13662-021-03615-2 | |
| 17. | Hüseyin Budak, Seda Kilinç Yildirim, Hasan Kara, Hüseyin Yildirim, On new generalized inequalities with some parameters for coordinated convex functions via generalized fractional integrals, 2021, 44, 0170-4214, 13069, 10.1002/mma.7610 | |
| 18. | Muhammad Aamir Ali, Hasan Kara, Jessada Tariboon, Suphawat Asawasamrit, Hüseyin Budak, Fatih Hezenci, Some New Simpson’s-Formula-Type Inequalities for Twice-Differentiable Convex Functions via Generalized Fractional Operators, 2021, 13, 2073-8994, 2249, 10.3390/sym13122249 | |
| 19. | Humaira Kalsoom, Hüseyin Budak, Hasan Kara, Muhammad Aamir Ali, Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals, 2021, 19, 2391-5455, 1153, 10.1515/math-2021-0072 | |
| 20. | İmdat İşcan, Erhan Set, Ahmet Ocak Akdemir, Alper Ekinci, Sinan Aslan, 2023, 9780323909532, 157, 10.1016/B978-0-32-390953-2.00017-7 | |
| 21. | MIAO-KUN WANG, SAIMA RASHID, YELIZ KARACA, DUMITRU BALEANU, YU-MING CHU, NEW MULTI-FUNCTIONAL APPROACH FOR κTH-ORDER DIFFERENTIABILITY GOVERNED BY FRACTIONAL CALCULUS VIA APPROXIMATELY GENERALIZED (ψ, ℏ)-CONVEX FUNCTIONS IN HILBERT SPACE, 2021, 29, 0218-348X, 2140019, 10.1142/S0218348X21400193 | |
| 22. | Saima Rashid, Dumitru Baleanu, Yu-Ming Chu, Some new extensions for fractional integral operator having exponential in the kernel and their applications in physical systems, 2020, 18, 2391-5471, 478, 10.1515/phys-2020-0114 | |
| 23. | Xuexiao You, Fatih Hezenci, Hüseyin Budak, Hasan Kara, New Simpson type inequalities for twice differentiable functions via generalized fractional integrals, 2022, 7, 2473-6988, 3959, 10.3934/math.2022218 | |
| 24. | YONG-MIN LI, SAIMA RASHID, ZAKIA HAMMOUCH, DUMITRU BALEANU, YU-MING CHU, NEW NEWTON’S TYPE ESTIMATES PERTAINING TO LOCAL FRACTIONAL INTEGRAL VIA GENERALIZED p-CONVEXITY WITH APPLICATIONS, 2021, 29, 0218-348X, 2140018, 10.1142/S0218348X21400181 | |
| 25. | Miguel Vivas-Cortez, Pshtiwan O. Mohammed, Y. S. Hamed, Artion Kashuri, Jorge E. Hernández, Jorge E. Macías-Díaz, On some generalized Raina-type fractional-order integral operators and related Chebyshev inequalities, 2022, 7, 2473-6988, 10256, 10.3934/math.2022571 | |
| 26. | Yu-Ming Chu, Saima Rashid, Fahd Jarad, Muhammad Aslam Noor, Humaira Kalsoom, More new results on integral inequalities for generalized $ \mathcal{K} $-fractional conformable Integral operators, 2021, 14, 1937-1179, 2119, 10.3934/dcdss.2021063 | |
| 27. | Nassima Nasri, Badreddine Meftah, Abdelkader Moumen, Hicham Saber, Fractional $ 3/8 $-Simpson type inequalities for differentiable convex functions, 2024, 9, 2473-6988, 5349, 10.3934/math.2024258 | |
| 28. | Fatih Hezenci, Hüseyin Budak, A note on fractional Simpson-like type inequalities for functions whose third derivatives are convex, 2023, 37, 0354-5180, 3715, 10.2298/FIL2312715H | |
| 29. | Zeynep Sanlı, Simpson type Katugampola fractional integral inequalities via Harmonic convex functions, 2022, 10, 23193786, 364, 10.26637/mjm1004/007 | |
| 30. | Meriem Merad, Badreddine Meftah, Abdelkader Moumen, Mohamed Bouye, Fractional Maclaurin-Type Inequalities for Multiplicatively Convex Functions, 2023, 7, 2504-3110, 879, 10.3390/fractalfract7120879 | |
| 31. | Fatih Hezenci, A Note on Fractional Simpson Type Inequalities for Twice Differentiable Functions, 2023, 73, 1337-2211, 675, 10.1515/ms-2023-0049 | |
| 32. | Hüseyin Budak, Fatih Hezenci, Hasan Kara, Mehmet Zeki Sarikaya, Bounds for the Error in Approximating a Fractional Integral by Simpson’s Rule, 2023, 11, 2227-7390, 2282, 10.3390/math11102282 | |
| 33. | Nazakat Nazeer, Ali Akgül, Faeem Ali, Study of the results of Hilbert transformation for some fractional derivatives, 2024, 0228-6203, 1, 10.1080/02286203.2024.2371685 | |
| 34. | Tarek Chiheb, Badreddine Meftah, Abdelkader Moumen, Mouataz Billah Mesmouli, Mohamed Bouye, Some Simpson-like Inequalities Involving the (s,m)-Preinvexity, 2023, 15, 2073-8994, 2178, 10.3390/sym15122178 | |
| 35. | Gamzenur Köksaldı, Tuba Tunç, Simpson-type inequalities for the functions with bounded second derivatives involving generalized fractional integrals, 2025, 2025, 1687-2770, 10.1186/s13661-025-02040-8 |
Figures(2)
Letizia Fracchia, Jareer J. Banat, Massimo Cavallo, Chiara Ceresa, Ibrahim M. Banat. Potential therapeutic applications of microbial surface-active compounds[J]. AIMS Bioengineering, 2015, 2(3): 144-162. doi: 10.3934/bioeng.2015.3.144
DownLoad: