Citation: Phuong Tran, Linh Nguyen, Huong Nguyen, Bong Nguyen, Linh Nong , Linh Mai, Huyen Tran, Thuy Nguyen, Hai Pham. Effects of inoculation sources on the enrichment and performance of anode bacterial consortia in sensor typed microbial fuel cells[J]. AIMS Bioengineering, 2016, 3(1): 60-74. doi: 10.3934/bioeng.2016.1.60
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In mathematical analysis and its applications, the hypergeometric function plays a vital role. Various special functions which are used in different branches of science are special cases of hypergeometric functions. Numerous extensions of special functions have introduced by many authors (see [1,2,3,4]). The generalized Gamma $ k $-function and its properties are broadly discussed in [5,6,7,8]. Later on, the researchers [9] motivated by the above idea and presented the $ k $-fractional integral and its applications. The integral representation of generalized confluent hypergeometric and hypergeometric $ k $-functions is presented by Mubeen and Habibullah [10]. The series solution of $ k $-hypergeometric differential equation is proposed by Mubeen et al. [11,12,13]. Li and Dong [14] established the hypergeometric series solutions for the second-order non-homogeneous $ k $-hypergeometric differential equation. The Generalized Wright $ k $-function and its different indispensable properties are briefly discussed in [15,16]. A class of Whittaker integral transforms involving confluent hypergeometric function and Fox H-function as kernels are discussed in [17,18]. An extension of some variant of Meijer type integrals in the class of Bohemians is elaborated in [19].
Generalization of Gr$ \ddot{u} $ss-type and Hermite-Hadamard type inequalities concerning with $ k $-fractional integrals are explored in [20,21]. Many researchers have derived the generalized forms of the Riemann-Liouville $ k $-fractional integrals and established several inequalities by considering various generalized fractional integrals. The readers may confer with [22,23,24,25,26] for details.
The Hadamard $ k $-fractional integrals and Hadamard-type inequalities for $ k $-fractional Riemann-Liouville integrals are presented by Farid et al. [27,28]. In [29,30], the authors have established inequalities by employing Hadamard-type inequalities for $ k $-fractional integrals. Nisar et al. [31] discussed Gronwall type inequalities by considering Riemann-Liouville $ k $ and Hadamard $ k $-fractional derivatives. Hadamard $ k $-fractional derivative and its properties are disscussed in [32]. Rahman et al. [33] described generalized $ k $-fractional derivative operator.
The Mittag-Leffler function naturally occurs in the solutions of fractional integrodifferential equations having an arbitrary order that is similar to that of the exponential function. The Mittag-Leffler functions have acquired significant recognition due to their wide applications in assorted fields. The Mittag-Leffler stability of fractional-order nonlinear dynamic systems is studied in [34]. Dos Santos discussed the Mittag-Leffler function in the diffusion process in [35]. The noteworthy role of the Mittag-Leffler function and its generalizations in fractional modelling is explored by Rogosin [36]. We suggest the readers to study the literature [37,38,39] for more details.
Now, we present some basic definitions and results.
The normalized Wright function is defined as
|
$ Wγ,δ(z)=Γ(δ)∞∑n=0znΓ(δ+nγ)n!,γ>−1,δ∈C. $
|
(1.1) |
Fox [40] and Wright [41] defined the Fox-Wright hypergeometric function $ _{u}\psi_{v} $ as:
|
$ uψv[(γ1,C1),.....,(γu,Cu)(δ1,D1)),.....,(δv,Dv)|z]=∞∑n=0∏ui=1Γ(γi+nCi)zn∏vi=1Γ(δi+nDi)n!, $
|
(1.2) |
where $ C_{i}\geq 0, i = 1, ..., u; D_{i}\geq 0, i = 1, ..., v. $ The function $ _{u}\psi_{v} $ is the generalized form of the well-known hypergeometric function $ _{u}F_{v} $ with u and v number of parameters in numerator and denominator respectively. It is given in [42] as:
|
$ uFv[γ1,.....,γuδ1,.....,δv|z]=∞∑n=0∏ui=1 (γi)nzn∏vi=1 (δi)nn!, $
|
(1.3) |
where $ (\kappa)_n $ is called the Pochhammer symbol and is defined by Petojevic [43] as:
| $ (\kappa)_0 = 1, \; and\; (\kappa)_n = \kappa (\kappa +1).....(\kappa +n-1) = \frac{\Gamma (\kappa +n)}{\Gamma (\kappa)}, n\in \mathbb{N}. $ |
From definitions (1.2) and (1.3), we have the following relation
|
$ uψv[(γ1,1),.....,(γu,1)(δ1,1),.....,(δv,1)|z]=Γ(γ1).....Γ(γu)Γ(δ1).....Γ(δv)uFv[γ1,.....,γuδ1,.....,δv|z]. $
|
(1.4) |
The Mittag-Leffler function with $ 2m $ parameters [44] is defined as follows:
|
$ E(δ,D)m(z)=∞∑n=0zn∏mi=1Γ(δi+nDi),z∈C $
|
(1.5) |
and in the form of Fox-Wright function it can be expressed as:
|
$ E(δ,D)m(z)=1ψm[(1,1)(δ1,D1),.....,(δv,Dv)|z],z∈C. $
|
(1.6) |
The function $ \phi_{(\gamma_1, \delta_2)}^{(\delta_1, D_1)} : (0, \infty)\rightarrow \mathbb{R} $ is defined in [45] as
|
$ ϕ(δ1,D1)(γ1,δ2)(z)=Γ(δ1)Γ(δ2)Γ(γ1)1ψ2[(γ1,1)(δ1,D1),(δ2,D2)|z],z∈C, $
|
(1.7) |
where $ \gamma_1, \delta_1, \delta_2 $ and $ D_1 > 0 $.
Diaz and Pariguan [46] proposed the following Pochhammer's $ k $-symbol $ (z)_{n, k} $ and the generalized Gamma $ k $-function by
|
$ (u)n,k=u(u+k)(u+2k)....(u+(n−1)k);u∈C,k∈R,n∈N+ $
|
|
$ Γk(u)=limn→∞=n!kn(nk)uk−1u)n,k,k>0,u∈C−kZ, $
|
respectively. They also introduced the generalized hypergeometric $ k $-function in [46] as follows:
|
$ uFv,k[(γ1,k),.....,(γu,k)(δ1,k),.....,(δv,k)|z]=∞∑n=0∏ui=1 (γi)n,k z n∏vi=1 (δi)n,k n!. $
|
(1.8) |
Now, we define the normalized Wright $ k $-function as follows:
| $ W_{\gamma, \delta, k}(z) = \Gamma_k(\delta) \sum\limits ^\infty_{n = 0}\frac{z^n}{\Gamma_k(\delta+ kn\gamma)n!}, \gamma > -1, \; \delta \in \mathbb{C} $ |
and the Fox -Wright type $ k $-function is as follows:
|
$ uψv[(γ1,kC1),.....,(γu,kCu)(δ1,kD1)),.....,(δv,kDv)|z]=∞∑n=0∏ui=1 Γk (γi+nkCi)zn∏vi=1 Γk (δi+nkDi)n!, $
|
(1.9) |
So from Eqs (1.9) and (1.8), we have
|
$ uψv,k[(γ1,k),.....,(γu,k)(δ1,k),.....,(δv,k)|z]=Γk(γ1).....Γk(γu)Γk(δ1).....Γk(δv)uFv,k[(γ1,k),.....,(γu,k)(δ1,k),.....,(δv,k)|z] $
|
(1.10) |
and (1.5) takes the form
|
$ E(δ,D)m,k(z)=∞∑n=0zn∏mi=1Γk(δi+nkDi),z∈C. $
|
(1.11) |
The function $ E_{(\delta, D)_{m, k}}(z) $ in terms of the function $ _{u}\psi_{v, k} $ is expressed as
|
$ E(δ,D)m,k(z)=1ψm,k[(1,k)(δ1,kD1),.....,(δv,kDv)|z]z∈C. $
|
(1.12) |
So the function (1.7) is given by
|
$ ϕ(δ1,D1)γ1,δ2,k(z)=Γk(δ1)Γk(δ2)Γk(γ1)1ψ2,k[(γ1,k)(δ1,kD1),(δ2,k)|z]z∈C. $
|
(1.13) |
Redheffer [47] presented the following inequality in 1969,
|
$ π2−x2π2+x2≤sinxx. $
|
(1.14) |
In the same year, Redheffer and Williams [48] proved the above inequality. Zhu and Sun [49] used the hyperbolic functions $ \sinh x $ and $ \cosh x $ and proved the following Redheffer-type inequalities
|
$ (r2+x2r2−x2)γ≤sinhxx≤(r2+x2r2−x2)δ $
|
(1.15) |
and
|
$ (r2+x2r2−x2)γ≤coshx≤(r2+x2r2−x2)δ1, $
|
(1.16) |
where $ 0 < x < r, \; \gamma \leq 0, \; \delta\geq\frac{r^{2}}{12} $ and $ \delta_{1}\geq \frac{r^{2}}{4} $. By using the inequalities (1.15) and (1.16), Mehrez proved the following Theorem (see[50]).
Theorem 1.1. The following inequalities
|
$ (s+zs−z)σγ,δ≤Wγ,δ(z)≤(s+zs−z)ργ,δ $
|
(1.17) |
holds true for all $ s > 0 $, $ \gamma, \delta > 0 $, $ 0 < z < s $, where $ \sigma_{\gamma, \delta} = 0 $ and $ \rho_{\gamma, \delta} = \frac{s\Gamma(\delta)}{2\Gamma(\delta+\gamma)} $ are the best possible constants and $ W_{\gamma, \delta} $ is the normalized Wright function defined by (1.1).
Recently, the same author has proved the following inequality (see[45]).
Theorem 1.2. Suppose that $ r, \gamma_1, \delta_1, \delta_2, D_1 > 0 $. Then the subsequent inequalities
|
$ (r+zr−z)λ(δ1 ,D1)γ1,δ2≤ϕ(δ1,D1)γ1,δ2(z)≤(r+zr−z)μ(δ1 ,D1)γ1,δ2 $
|
(1.18) |
are true for all $ z \in (0, r) $, where $ \lambda ^{(\delta_1, D_1)}_{\gamma_1, \delta_2} = 0 $ and $ \mu ^{(\delta_1, D_1)}_{\gamma_1, \delta_2} = \frac{r \gamma_1\Gamma(\delta_1)}{2\delta_2\Gamma(\delta_1+D_1)} $ are the best possible constants. Where $ \phi_{\gamma_1, \delta_2}^{(\delta_1, D_1)}(z) $ is defined in (1.7).
In this article, we extend the Redheffer-type inequality (1.18) for the normalized Fox-Wright $ k $-functions $ \phi_{\gamma_1, \delta_2, k}^{(\delta_1, D_1)}(z) $ and establish new Redheffertype inequalities for the hypergeometric $ k $-function $ _{1}F_{2, k} $ and for the four parametric Mittag-Leffler $ k $-function $ \tilde{E}_{\delta_1, D_1;\delta_2, 1, k}(z) = \Gamma_k(\delta_1)\Gamma_k(\delta_2)E_{\delta_1, D_1;\delta_2, 1, k}(z) $.
In this section, we first state the two lemmas which are helpful for proving the main results. The first lemma from [51] describes the monotonicity of two power series, and the second lemma is the monotone form of the L'Hospitals' rule [52]. The two lemmas are stated below.
Lemma 2.1. Suppose that the two sequences $ \{u_n\}_{n\geq 0} $ and $ \{v_n\}_{n\geq 0} $ of real numbers, and let the power series $ f(x) = \sum_{n\geq0}u_n x^n $ and $ g(x) = \sum_{n\geq0}v_n x^n $ be convergent for $ |x| < r $. If $ v_n > 0 $ for $ n\geq 0 $ and if the sequence $ \{\frac{u_n}{v_n}\}_{n\geq 0} $ is (strictly) increasing (decreasing), then the function $ x\rightarrow \frac{f(x)}{g(x)} $ is (strictly) increasing (decreasing) on $ (0, r) $.
Lemma 2.2. Suppose that the two functions $ f_1, g_1 : [a, b]\rightarrow R $ be continuous and differentiable on $ (a, b) $. Assume that $ g_1^\prime\neq 0 $ on $ (a, b) $. If $ \frac{f_1^\prime}{g_1^\prime} $ is increasing (decreasing) on $ (a, b) $, then the functions
| $ x\mapsto \frac{f_1(x)-f_1(a)}{g_1(x)-g_1(a)}\; and \; x\mapsto \frac{f_1(x)-f_1(b)}{g_1(x)-g_1(b)} $ |
are also increasing (decreasing) on $ (a, b) $.
Theorem 2.3. Let$ r > 0 \gamma_1, \delta_1, \delta_2, D_1 > 0 $. Then the following inequalities holds
|
$ (r+zr−z)λ(δ1 ,D1)γ1,δ2≤ϕ(δ1 ,D1)γ1,δ2,k(z)≤(r+zr−z)μ(δ1 ,D1)γ1,δ2, $
|
(2.1) |
where $ z \in (0, r) $, $ \lambda ^{(\delta_1, D_1)}_{\gamma_1, \delta_2} = 0 $ and $ \mu ^{(\delta_1, D_1)}_{\gamma_1, \delta_2} = \frac{r \gamma_1\Gamma_k(\delta_1)}{2\delta_2\Gamma_k(\delta_1+kD_1)} $ are the appropriate constants.
Proof. From definition of the function $ \phi_{\gamma_1, \delta_2, k}^{(\delta_1, D_1)}(z) $ from Eq (1.13), we have
|
$ ddz(ϕ(δ1 ,D1)γ1,δ2,k(z))=Γk(δ1)Γk(δ2)Γk(γ1)×∞∑n=0Γk(γ1+(n+1)k)znΓk(δ1+D1(n+1)k)Γk(δ2+(n+1)k)n!. $
|
(2.2) |
Let
| $ M(z) = \frac{\log \phi_{\gamma_{\ \ 1} \ \ , \delta_{\ \ 2} \ \ , k}^{(\delta_{\ \ 1}\;, \ \ D_{\ \ 1}\;)} \ \ (z)}{\log (\frac{r+z}{r-z}\;)} = \frac{g(z)}{h(z)}, $ |
where $ g(z) = \log (\phi_{\gamma_1, \delta_2, k}^{(\delta_1, D_1)}(z)) $ and $ h(z) = \log (\frac{r+z}{r-z}) $. Now we will find $ \frac{g'(z)}{h'(z)} $ as given below
| $ \frac{g'(z)}{h'(z)} = \frac{(r^{2}-z^{2})\frac{d}{dz}(\phi_{\gamma_{\ \ 1} \ \ , \delta_{\ \ 2} \ \ , k}^{(\delta_{\ \ 1}, \ \ D_{\ \ 1})} \ \ (z))}{2r \phi_{\gamma_{\ \ 1} \ \ , \delta_{\ \ 2} \ \ , k}^{(\delta_{\ \ 1} \ \ , D_{\ \ 1})} \ \ (z)} = \frac{G(z)}{2rH(z)}, $ |
where $ G(z) = (r^{2}-z^{2})\frac{d}{dz}(\phi_{\gamma_1, \delta_2, k}^{(\delta_1, D_1)}) $ and $ H(z) = \phi_{\gamma_1, \delta_2, k}^{(\delta_1, D_1)}(z) $
Using 2.2, we have
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$ G(z)=Γk(δ1)Γk(δ2)Γk(γ1)(r2−z2)×∞∑n=0Γk(γ1+(n+1)k)znΓk(δ1+D1(n+1)k)Γk(δ2+(n+1)k)n!=Γk(δ1)Γk(δ2)Γk(γ1)(r2−z2)(∞∑n=0Γk(γ1+(n+1)k)znΓk(δ1+D1(n+1)k)Γk(δ2+(n+1)k)n!−∞∑n=2Γk(γ1+(n−1)k)znΓk(δ1+D1(n−1)k)Γk(δ2+(n−1)k)(n−2)!)=r2Γk(δ1)Γk(δ2)Γk(γ1+k)Γk(γ1)Γk(δ1+D1k)Γk(δ2+k)+r2Γk(δ1)Γk(δ2)Γk(γ1+2k)Γk(γ1)Γk(δ1+2D1k)Γk(δ2+2k)+Γk(δ1)Γk(δ2)Γk(γ1)∞∑n=2(r2Γk(γ1+(n+1)k)znΓk(δ1+D1(n−+)k)Γk(δ2+(n+1)k)n!−Γk(γ1+(n−1)k)znΓk(δ1+D1(n−1)k)Γk(δ2+(n−1)k)(n−2)!)zn=∞∑n=0vnzn, $
|
where $ b_0 $, $ b_1 $ and $ v_n $ are as follows:
| $ b_0 = \frac{r^2\Gamma_k(\delta_1)\Gamma_k(\delta_2)\Gamma_k(\gamma_1+ k)}{\Gamma_k(\gamma_1)\Gamma_k(\delta_1+ D_1k)\Gamma_k(\delta_2+ k)}, \; \; b_1 = \frac{r^2\Gamma_k(\delta_1)\Gamma_k(\delta_2)\Gamma_k(\gamma_1+ 2k)}{\Gamma_k(\gamma_1)\Gamma_k(\delta_1+ 2D_1k)\Gamma_k(\delta_2+ 2k)} $ |
and for $ n\geq 2 $, we have
| $ v_n = \frac{\Gamma_k(\delta_1)\Gamma_k(\delta_2)}{\Gamma_k(\gamma_1)}\sum\limits ^\infty_{n = 2}\Bigl(\frac{r^2\Gamma_k(\gamma_1+(n+1)k)} {\Gamma_k(\delta_1 +D_1 (n-+)k)\Gamma_k(\delta_2+(n+1)k)n!} $ |
| $ -\frac{\Gamma_k(\gamma_1+(n-1)k)z^{n}}{\Gamma_k(\delta_1 +D_1 (n-1)k)\Gamma_k(\delta_2+(n-1)k)(n-2)!}\Bigr). $ |
Similarly, we can write $ H(z) $ as
| $ H(z) = \sum\limits ^\infty_{n = 0}d_n z^n, $ |
where
| $ d_n = \frac{\Gamma_k(\delta_1)\Gamma_k(\delta_2)\Gamma_k(\gamma_1+ nk)}{\Gamma_k(\gamma_1)\Gamma_k(\delta_1+ nkD_1)\Gamma_k(\delta_2+ nk)}. $ |
Next, we consider the sequence $ w_n = \frac{v_n}{d_n} $ such that $ w_0 = a_0 $ and $ w_1 = \frac{b_1}{d_1} $ and so on.
Now, we have
|
$ w1−w0=r2Γk(δ2+k)Γk(γ1+2k)Γk(δ1+kD1)Γk(δ2+2k)Γk(δ1+2kD1)Γk(γ1+k)−r2Γk(δ1)Γk(δ2)Γk(γ1+k)Γk(γ1)Γk(δ1+D1k)Γk(δ2+k)=r2(γ1+k)Γk(δ1+kD1)(δ2+k)Γk(δ1+2D1k)−r2Γk(δ1)δ2)Γk(δ1+D1k)≤γ1δ2[Γk(δ1+kD1))Γk(δ1+2kD1)−Γk(δ1)Γk(δ1+kD1)]≤0. $
|
(2.3) |
Since $ \gamma_1\geq \delta_2 $ and employing the log-convexity property of $ \Gamma_k $ function, the ratio $ z \mapsto \frac{\Gamma_k(z+c)}{\Gamma_k(z)} $ is increasing on $ (0, \infty) $ for $ c > 0 $. This shows that the inequality stated below
|
$ Γk(z+c)Γk(z)≤Γk(z+c+d)Γk(z+d) $
|
(2.4) |
holds for all $ z, c, d > 0 $. Using (2.4) in (2.3) by letting $ z = \delta_1 $ and $ c = d = kD_1 $, we observed that $ w_1\leq w_0 $. Likely, we have
|
$ w2−w1=r2Γk(γ1+3k)Γk(δ2+2k)Γk(δ1+2kD1)Γk(δ2+3k)Γk(δ1+3kD1)Γk(γ1+2k)−r2Γk(δ2+k)Γk(δ1+kD1)Γk(γ1+2k)Γk(δ2+2k)Γk(δ1+2kD1)Γk(γ1+k)−2Γk(δ2+2k)Γk(δ1+2kD1)Γk(γ1+k)Γk(δ2+k)Γk(δ1+kD1)Γk(γ1+2k)=r2(γ1+2k)Γk(γ1+2k)(δ2+k)Γk(δ2+k)Γk(δ1+2kD1)(δ2+2k)Γk(δ2+2k)Γk(δ1+3kD1)(γ1+k)Γk(γ1+k)−r2Γk(δ2+k)Γk(δ1+kD1)Γk(γ1+2k)Γk(δ2+2k)Γk(δ1+2kD1)Γk(γ1+k)−2Γk(δ2+2k)Γk(δ1+2kD1)Γk(γ1+k)Γk(δ2+k)Γk(δ1+kD1)Γk(γ1+2k) $
|
|
$ =r2Γk(γ1+2k)Γk(δ2+k)Γk(δ2+2k)Γk(γ1+k)[(γ1+2k)(δ2+k)Γk(δ1+2kD1)(γ1+k)(δ2+2k)Γk(δ1+3kD1)−Γk(δ1+kD1)Γk(δ1+2kD1)]−2Γk(δ2+2k)Γk(δ1+2kD1)Γk(γ1+k)Γk(δ2+k)Γk(δ1+kD1)Γk(γ1+2k)≤0. $
|
(2.5) |
Since by using Eq (2.4), when $ z = \delta_1+kD_1 $ and $ c = d = kD_1 $, we have $ w_2\leq w_1. $
Now, for $ n\geq 2 $, we have
|
$ wn+1−wn=r2Γk(γ1+(n+2)k)Γk(δ2+(n+1)k)Γk(δ1+(n+1)kD1)Γk(γ1+(n+1)k)Γk(δ2+(n+1)k)Γk(δ1+(n+2)kD1)−(n+1)!Γk(γ1+nk)Γk(δ2+(n+1)k)Γk(δ1+(n+1)kD1)(n−1)!Γk(γ1+(n+1)k)Γk(δ2+nk)Γk(δ1+nkD1)−r2Γk(γ1+(n+1)k)Γk(δ2+nk)Γk(δ1+nkD1)Γk(γ1+nk)Γk(δ2+(n+1)k)Γk(δ1+(n+1)kD1)+n!Γk(γ1+(n−1)k)Γk(δ2+nk)Γk(δ1+nkD1)(n−2)!Γk(γ1+nk)Γk(δ2+(n11)k)Γk(δ1+(n−1)kD1)=r2[(γ1+(n+1)k)Γk(δ1+(n+1)kD1)(δ2+(n+1)k)Γk(δ1+(n+2)kD1)−(γ1+n)k)Γk(δ1+nkD1)(δ2+nk)Γk(δ1+(n+1)kD1)]+n!(n−2)![(δ2+(n−1)k)Γk(δ1+nkD1)(γ1+(n−1)k)Γk(δ1+(n−2)kD1)−(n+1)(δ2+n)k)Γk(δ1+(n+1)kD1)(n−1)(γ1+nk)Γk(δ1+nkD1)]≤r2(γ1+nk)(δ2+nk)[Γk(δ1+(n+1)kD1)Γk(δ1+(n+2)kD1)−Γk(δ1+nkD1)Γk(δ1+(n+1)kD1)]+n!(δ2+(n−1)k)(n−2)!(γ+(n−1)k)[Γk(δ1+nkD1)Γk(δ1+(n−1)kD1)−Γk(δ1+(n+1)kD1)Γk(δ1+nkD1)] $
|
(2.6) |
Using $ z = \delta_1 +nkD_1 $ and $ c = d = kD_1 $, we have the inequality (2.4) in the following form
|
$ Γk(δ1+(n+2)kD1)Γk(δ1+nkD1)−Γ2k(δ1+(n+1)kD1)≥0 $
|
(2.7) |
and similarly for using $ z = \delta_1 +(n-1)kD_1 $ and $ c = d = kD_1 $, we have from (2.4)
|
$ Γk(δ1+(n+2)kD1)Γk(δ1+nkD1)−Γ2k(δ1+(n+1)kD1)≥0 $
|
(2.8) |
so by using (2.7) and (2.8) in (2.6), we have
|
$ wn+1≤wn. $
|
(2.9) |
Now from Eqs (2.3), (2.10) and (2.9), we conclude that the sequence $ \{w_n\}_{n\geq0} $ is a decreasing sequence.
From Lemma (2.1), we deduce that $ \frac{g'}{h'} $ is decreasing on $ (0, r) $ and accordingly the function $ M(z) $ is also decreasing on $ (0, r) $. Alternatively, using the Bernoulli-L'Hospital's rule we get
| $ \lim\limits_{z\rightarrow0}M(z) = \frac{r\gamma_1\Gamma_k(\delta_1)}{2\delta_2\Gamma_k(\delta_1+kD_1)} \; and\; \lim\limits_{z\rightarrow0}M(z) = 0. $ |
It is essential to reveal that there is another proof of this theorem which is described as following.
For this, we define a function $ \Upsilon : (0, r)\rightarrow \mathbb {R} $ by
| $ \Upsilon(z) = \frac{r\gamma_1\Gamma_k(\delta_1)}{2\delta_2\Gamma_k(\delta_1+D_1)}\log(\frac{r+z}{r-z})-\log (\phi_{\gamma_1, \delta_2, k}^{(\delta_1, kD_1)}(z)). $ |
Consequently,
|
$ ϕ(δ1,kD1)γ1,δ2,k(z)Υ′(z)=r2γ1Γk(δ1)δ2Γk(δ1+D1)(r2−z2)ϕ(δ1,D1)γ1,δ2,k(z)−ddz(ϕ(δ1,D1)γ1,δ2,k(z))=Γk(δ1)Γk(δ2)Γk(γ1)[r2γ1Γk(δ1)δ2Γk(δ1+kD1)(r2−z2)∞∑n=0Γk(γ1+nk)znΓk(δ1+D1nk)Γk(δ2+nk)n!−∞∑n=0Γk(γ1+(n+1)k)znΓk(δ1+D1(n+1)k)Γk(δ2+(n+1)k)n!]≥Γk(δ1)Γk(δ2)Γk(γ1)[γ1Γk(δ1)δ2Γk(δ1+kD1)∞∑n=0Γk(γ1+nk)znΓk(δ1+D1nk)Γk(δ2+nk)n!−∞∑n=0Γk(γ1+(n+1)k)znΓk(δ1+D1(n+1)k)Γk(δ2+(n+1)k)n!]=Γk(δ1)Γk(δ2)Γk(γ1)∞∑n=0Γk(γ1+nk)znΓk(δ2+nk)n![γ1Γk(δ1)δ2Γk(δ1+kD1)−(γ1+nk)(δ2+nk)Γk(δ1+D1(n+1)k)]. $
|
(2.10) |
Since $ \gamma_1\geq\delta_2 $, therefore $ \frac{\gamma_1}{\delta_2}\geq \frac{\gamma_1+nk}{\delta_2+nk} $ for every $ n\geq 0 $ and resultantly
|
$ ϕ(δ1,D1)γ1,δ2,k(z)Υ′(z)≥Γk(δ1)Γk(δ2)Γk(γ1)∞∑n=0Γk(γ1+nk)znΓk(δ2+nk)n![Γk(δ1)Γk(δ1+kD1)−1Γk(δ1+kD1(n+1))]. $
|
Now, using the values $ z = \delta_1, \; c = kD_1 $ and $ d = nkD_1 $ in (2.4), function $ \Upsilon(z) $ is increasing on $ (0, r) $ and consequently $ \Upsilon(z)\geq\Upsilon(0) = 0 $. This completes the proof of right hand side of the inequality 2.1. For proving the left hand side of (2.1), we conclude from Eq (2.2) that the function $ \phi_{\gamma_1, \delta_2, k}^{(\delta_1, D_1)}(z) $ is increasing on $ (0, \infty) $ and finally
| $ \phi_{\gamma_1, \delta_2, k}^{(\delta_1, D_1)}(z)\geq \phi_{\gamma_1, \delta_2, k}^{(\delta_1, D_1)}(0) = 1. $ |
This ends the proof process of theorem $ 2.3. $
Corollary 2.4. Let $ r, \gamma_1, \delta_1, \delta_2 > 0 $. Then inequalities
|
$ (r+zr−z)λ(δ1,1)γ1,δ2≤1F2,k(γ1;δ1,δ2;z)≤(r+zr−z)μ(δ1,1)γ1,δ2, $
|
are true for all $ z \in (0, r) $, where $ \lambda ^{(\delta_1, 1)}_{\gamma_1, \delta_2} = 0 $ and $ \mu ^{(\delta_1, 1)}_{\gamma_1, \delta_2} = \frac{r \gamma_1}{2\delta_2\delta_1} $ are the suitable values of constants.
Proof. This inequality can be proved by using $ D_1 = 1 $ in (2.1).
Corollary 2.5. Let $ r, \delta_1, D_1 > 0 $. If $ 0 < \delta_2\leq 1, $ then the following inequalities
| $ (\frac{r+z}{r-z})^{\lambda ^{(\delta_1, D_1)}_{1, \delta_2}}\leq \tilde{E}_{\delta_1, D_1;\delta_2, 1, k}(z) \leq (\frac{r+z}{r-z})^{\mu ^{(\delta_1, D_1)}_{1, \delta_2}}, $ |
hold for all $ z \in (0, r) $, where $ \lambda ^{(\delta_1, D_1)}_{1, \delta_2} = 0 $ and $ \mu ^{(\delta_1, D_1)}_{1, \delta_2} = \frac{r \Gamma_k(\delta_1)}{2\delta_2\Gamma_k(\delta_1+kD_1)} $ are the suitable values of constants.
Proof. This inequality can be proved by using $ \gamma_1 = 1 $ in (2.1).
Remark 2.6. The particular cases of the inequalities in Corollaries 2.4 and 2.5 for $ k = 1 $ are reduce to the results in [45,Corollaries 1.2 and 1.3] respectively.
This article deals with the Redheffer-type inequalities by using the more general Fox-Wright function. Moreover, from the newly established inequalities, the results for generalized hypergeometric functions and the four-parametric generalized Mittag-Leffler functions are also obtained by using the appropriate values of exponents in generalized inequalities. The obtained results are more general, as shown by relating the special cases with the existing literature. The idea we used in this article attracts scientists' attention, and they may stimulate further research in this direction.
Taif University Researchers Supporting Project number (TURSP-2020/275), Taif University, Taif, Saudi Arabia.
The authors declares that there is no conflict of interests regarding the publication of this paper.
| [1] |
Ha PT, Tae B, Chang IS (2008) Performance and bacterial consortium of microbial fuel cell fed with formate. Energ Fuels 22: 164–168. doi: 10.1021/ef700294x
|
| [2] |
Du Z, Li H, Gu T (2007) A state of the art review on microbial fuel cells: A promising technology for wastewater treatment and bioenergy. Biotechnol Adv 25: 464–482. doi: 10.1016/j.biotechadv.2007.05.004
|
| [3] |
Logan BE (2006) Microbial fuel cells: methodology and technology. Environ Sci Technol 40: 5181–5192. doi: 10.1021/es0605016
|
| [4] | Kim HJ, Hyun MS, Chang IS, et al. (1999) A microbial fuel cell type lactate biosensor using a metal-reducing bacterium, Shewanella putrefaciens. J Microbiol Biotechn 9: 365–367. |
| [5] |
Kim BH, Chang IS, Gil GC, et al. (2003) Novel BOD (biological oxygen demand) sensor using mediator-less microbial fuel cell. Biotechnol Lett 25: 541–545. doi: 10.1023/A:1022891231369
|
| [6] |
Chang IS, Jang JK, Gil GC, et al. (2004) Continuous determination of biochemical oxygen demand using microbial fuel cell type biosensor. Biosens Bioelectron 19: 607–613. doi: 10.1016/S0956-5663(03)00272-0
|
| [7] |
Gil GC, Chang IS, Kim BH, et al. (2003) Operational parameters affecting the performannce of a mediator-less microbial fuel cell. Biosens Bioelectron 18: 327–334. doi: 10.1016/S0956-5663(02)00110-0
|
| [8] |
Moon H, Chang IS, Kang KH, et al. (2004) Improving the dynamic response of a mediator-less microbial fuel cell as a biochemical oxygen demand (BOD) sensor. Biotechnol Lett 26: 1717–1721. doi: 10.1007/s10529-004-3743-5
|
| [9] |
Kim M, Hyun MS, Gadd GM, et al. (2007) A novel biomonitoring system using microbial fuel cells. J Environ Monitor 9: 1323–1328. doi: 10.1039/b713114c
|
| [10] |
van der Schalie WH, Shedd TR, Knechtges PL, et al. (2001) Using higher organisms in biological early warning systems for real-time toxicity detection. Biosens Bioelectron 16: 457–465. doi: 10.1016/S0956-5663(01)00160-9
|
| [11] |
Ren Z, Zha J, Ma M, et al. (2007) The early warning of aquatic organophosphorus pesticide contamination by on-line monitoring behavioral changes of Daphnia magna. Environ Monit Assess 134: 373–383. doi: 10.1007/s10661-007-9629-y
|
| [12] |
Pham TH, Boon N, Aelterman P, et al. (2008) High shear enrichment improve the performance of the anodophillic microbial consortium in a microbial fuel cell. Microb Biotechnol 1: 487–496. doi: 10.1111/j.1751-7915.2008.00049.x
|
| [13] |
Pham H, Boon N, Marzorati M, et al. (2009) Enhanced removal of 1,2-dichloroethane by anodophilic microbial consortia. Water Res 43: 2936–2946. doi: 10.1016/j.watres.2009.04.004
|
| [14] |
Logan BE, Regan JM (2006) Electricity-producing bacterial communities in microbial fuel cells. Trends Microbiol 14: 512–518. doi: 10.1016/j.tim.2006.10.003
|
| [15] |
Rabaey K, Boon N, Siciliano SD, et al. (2004) Biofuel cells select for microbial consortia that self-mediate electron transfer. Appl Environ Microbiol 70: 5373–5382. doi: 10.1128/AEM.70.9.5373-5382.2004
|
| [16] |
Vázquez-Larios AL, Poggi-Varaldo HM, Solorza-Feria O, et al. (2015) Effect of type of inoculum on microbial fuel cell performance that used RuxMoySez as cathodic catalyst. Int J Hydrogen Energ 40: 17402–17412. doi: 10.1016/j.ijhydene.2015.09.143
|
| [17] |
Ortega-Martínez AC, Juárez-López K, Solorza-Feria O, et al. (2013) Analysis of microbial diversity of inocula used in a five-face parallelepiped and standard microbial fuel cells. Int J Hydrogen Energ 38: 12589–12599. doi: 10.1016/j.ijhydene.2013.02.023
|
| [18] |
Li XM, Cheng KY, Selvam A, et al. (2013) Bioelectricity production from acidic food waste leachate using microbial fuel cells: Effect of microbial inocula. Process Biochem 48: 283–288. doi: 10.1016/j.procbio.2012.10.001
|
| [19] | Greenberg A, Clesceri LS, Eaton AD (1992) Standard Methods for the Examination of Water and Wastewater, 18th Eds., Washington: American Public Health Association. |
| [20] |
Boon N, De Gelder L, Lievens H, et al. (2002) Bioaugmenting bioreactors for the continuous removal of 3-chloroaniline by a slow release approach. Environ Sci Technol 36: 4698–4704. doi: 10.1021/es020076q
|
| [21] | Muyzer G, de Waal EC, Uitterlinden AG (1993) Profiling of complex microbial populations by denaturing gradient gel electrophoresis analysis of polymerase chain reaction-amplified genes coding for 16S rRNA. Appl Environ Microbiol 59: 695–700. |
| [22] | Boon N, De Windt W, Verstraete W, et al. (2002) Evaluation of nested PCR-DGGE (denaturing gradient gel electrophoresis) with group-specific 16S rRNA primers for the analysis of bacterial communities from different wastewater treatment plants. FEMS Microbiol Ecol 39: 101–112. |
| [23] |
Altschul SF, Gish W., Miller W, et al. (1990) Basic local alignment search tool. J Mol Biol 215: 403–410. doi: 10.1016/S0022-2836(05)80360-2
|
| [24] |
Kim JR, Beecroft NJ, Varcoe JR, et al. (2011) Spatiotemporal development of the bacterial community in a tubular longitudinal microbial fuel cell. Appl Microbiol Biotechnol 90: 1179–1191. doi: 10.1007/s00253-011-3181-y
|
| [25] | Stein NE, Hamelers HVM, Buisman CNJ (2012) The effect of different control mechanisms on the sensitivity and recovery time of a microbial fuel cell based biosensor. Sensors and Actuators B: Chemical 171–172: 816–821. |
| [26] |
Lower SK, Hochella MF, Beveridge TJ (2001) Bacterial recognition of mineral surfaces: nanoscale interactions between Shewanella and alpha-FeOOH. Science 292: 1360–1363. doi: 10.1126/science.1059567
|
| [27] |
Logan BE (2009) Exoelectrogenic bacteria that power microbial fuel cells. Nat Rev Micro 7: 375–381. doi: 10.1038/nrmicro2113
|
| [28] |
Bond DR, Lovley DR (2005) Evidence for involvement of an electron shuttle in electricity generation by Geothrix fermentans. Appl Environ Microbiol 71: 2186–2189. doi: 10.1128/AEM.71.4.2186-2189.2005
|
| [29] |
Huang JS, Guo Y, Yang P, et al. (2014) Performance evaluation and bacteria analysis of AFB-MFC enriched with high-strength synthetic wastewater. Water Sci Technol 69: 9–14. doi: 10.2166/wst.2013.390
|
| [30] |
Rabaey K, Rodriguez J, Blackall LL, et al. (2007) Microbial ecology meets electrochemistry: electricity-driven and driving communities. ISME J 1: 9–18. doi: 10.1038/ismej.2007.4
|
| [31] |
Vázquez-Larios AL, Solorza-Feria O, Vázquez-Huerta G, et al. (2011) Effects of architectural changes and inoculum type on internal resistance of a microbial fuel cell designed for the treatment of leachates from the dark hydrogenogenic fermentation of organic solid wastes. Int J Hydrogen Energ 36: 6199–6209. doi: 10.1016/j.ijhydene.2011.01.006
|
| [32] |
Kim GT, Webster G, Wimpenny JW, et al. (2006) Bacterial community structure, compartmentalization and activity in a microbial fuel cell. J Appl Microbiol 101: 698–710. doi: 10.1111/j.1365-2672.2006.02923.x
|
| [33] | Madigan MT, Martinko J, Parker J (2004) Brock Biology of Microorganisms. NJ: Pearson Education Inc. 991. |
| [34] |
Rabaey K, Verstraete W (2005) Microbial fuel cells: novel biotechnology for energy generation. Trends Biotechnol 23: 291–298. doi: 10.1016/j.tibtech.2005.04.008
|
| [35] |
Pham TH, Boon N, De Maeyer K, et al. (2008) Use of Pseudomonas species producing phenazine-based metabolites in the anodes of microbial fuel cells to improve electricity generation. Appl Microbiol Biotechnol 80: 985–993. doi: 10.1007/s00253-008-1619-7
|
| [36] |
Rabaey K, Boon N, Hofte M, et al. (2005) Microbial phenazine production enhances electron transfer in biofuel cells. Environ Sci Technol 39: 3401–3408. doi: 10.1021/es048563o
|
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Figures(4)
Phuong Tran, Linh Nguyen, Huong Nguyen, Bong Nguyen, Linh Nong , Linh Mai, Huyen Tran, Thuy Nguyen, Hai Pham. Effects of inoculation sources on the enrichment and performance of anode bacterial consortia in sensor typed microbial fuel cells[J]. AIMS Bioengineering, 2016, 3(1): 60-74. doi: 10.3934/bioeng.2016.1.60
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