Citation: Mohammad Azizur Rahman, Shahdat Hossain, Noorlidah Abdullah, Norhaniza Aminudin. Brain proteomics links oxidative stress with metabolic and cellular stress response proteins in behavioural alteration of Alzheimer’s disease model rats[J]. AIMS Neuroscience, 2019, 6(4): 299-315. doi: 10.3934/Neuroscience.2019.4.299
| [1] | Ghulam Farid, Hafsa Yasmeen, Hijaz Ahmad, Chahn Yong Jung . Riemann-Liouville Fractional integral operators with respect to increasing functions and strongly $ (\alpha, m) $-convex functions. AIMS Mathematics, 2021, 6(10): 11403-11424. doi: 10.3934/math.2021661 |
| [2] | Yu-Pei Lv, Ghulam Farid, Hafsa Yasmeen, Waqas Nazeer, Chahn Yong Jung . Generalization of some fractional versions of Hadamard inequalities via exponentially $ (\alpha, h-m) $-convex functions. AIMS Mathematics, 2021, 6(8): 8978-8999. doi: 10.3934/math.2021521 |
| [3] | Shuang-Shuang Zhou, Ghulam Farid, Chahn Yong Jung . Convexity with respect to strictly monotone function and Riemann-Liouville fractional Fejér-Hadamard inequalities. AIMS Mathematics, 2021, 6(7): 6975-6985. doi: 10.3934/math.2021409 |
| [4] | Hengxiao Qi, Muhammad Yussouf, Sajid Mehmood, Yu-Ming Chu, Ghulam Farid . Fractional integral versions of Hermite-Hadamard type inequality for generalized exponentially convexity. AIMS Mathematics, 2020, 5(6): 6030-6042. doi: 10.3934/math.2020386 |
| [5] | Moquddsa Zahra, Muhammad Ashraf, Ghulam Farid, Kamsing Nonlaopon . Inequalities for unified integral operators of generalized refined convex functions. AIMS Mathematics, 2022, 7(4): 6218-6233. doi: 10.3934/math.2022346 |
| [6] | Miguel Vivas-Cortez, Muhammad Aamir Ali, Artion Kashuri, Hüseyin Budak . Generalizations of fractional Hermite-Hadamard-Mercer like inequalities for convex functions. AIMS Mathematics, 2021, 6(9): 9397-9421. doi: 10.3934/math.2021546 |
| [7] | Jamshed Nasir, Shahid Qaisar, Saad Ihsan Butt, Hassen Aydi, Manuel De la Sen . Hermite-Hadamard like inequalities for fractional integral operator via convexity and quasi-convexity with their applications. AIMS Mathematics, 2022, 7(3): 3418-3439. doi: 10.3934/math.2022190 |
| [8] | Ghulam Farid, Maja Andrić, Maryam Saddiqa, Josip Pečarić, Chahn Yong Jung . Refinement and corrigendum of bounds of fractional integral operators containing Mittag-Leffler functions. AIMS Mathematics, 2020, 5(6): 7332-7349. doi: 10.3934/math.2020469 |
| [9] | Manar A. Alqudah, Artion Kashuri, Pshtiwan Othman Mohammed, Muhammad Raees, Thabet Abdeljawad, Matloob Anwar, Y. S. Hamed . On modified convex interval valued functions and related inclusions via the interval valued generalized fractional integrals in extended interval space. AIMS Mathematics, 2021, 6(5): 4638-4663. doi: 10.3934/math.2021273 |
| [10] | Atiq Ur Rehman, Ghulam Farid, Sidra Bibi, Chahn Yong Jung, Shin Min Kang . $k$-fractional integral inequalities of Hadamard type for exponentially $(s, m)$-convex functions. AIMS Mathematics, 2021, 6(1): 882-892. doi: 10.3934/math.2021052 |
A function plays very vital role in studying real analysis, numerical analysis, functional analysis and statistical analysis etc. Properties of a function like continuity, differentiability increase its importance in the subjects of mathematical analysis, economics, differential equations etc. A convex function defined about a century ago has fascinating geometric and analytical properties, due to which it is studied frequently in the subjects of mathematical analysis and optimization theory. Its extended form is the well-known Jensen inequality which in particular generates several classical inequalities. In the development of the theory of mathematical inequalities its role is significant. Description of a convex function in different convenient forms motivates the researchers to extend its concept in the form of new definitions and notions. Especially, to generalize and refine the classical inequalities for convex functions, it can be found in the literature that researchers have defined many new classes of functions.
In the theory of inequalities the class of convex functions has been considered at very large scale in past few decades, see [1,2,3,4,5] and references therein.
A function $ f $ defined on an interval $ I $ is called convex if the following inequality is satisfied [6]:
|
$ f(tx+(1−t)y)≤tf(x)+(1−t)f(y),t∈[0,1],x,y∈ I. $
|
(1.1) |
A function $ f $ is said to be convex with respect to a strictly monotone function $ g $ if $ f\circ g^{-1} $ is a convex function, see [7,Definition 1.19,p.7]. An analytic representation (1.1) of convex function provides the motivation to define new classes of functions. By including some convenient parameters and functions, various numerous generalizations, extensions and refinements have been introduced. For example in [8,9,10,11,12], authors have defined $ m $-convex, $ h $-convex, $ s $-convex, Godunova-Levin convex, Godunova-Levin $ s $-convex, $ (\alpha, m) $-convex, $ (h-m) $-convex, $ (s, m) $-convex and $ P $-convex functions. All such functions can be reproduced from the following class of functions called $ (\alpha, h-m) $-convex functions.
Let $ J\subseteq\mathbb{R} $ be an interval containing $ (0, 1) $ and let $ h:J\rightarrow \mathbb{R} $ be a non-negative function, $ h\not\equiv0 $. A function $ f:[0, b]\rightarrow \mathbb{R} $ is said to be $ (\alpha, h-m) $-convex, if $ f $ is non-negative and for all $ x, y\in [0, b] $, $ (\alpha, m)\in[0, 1]^{2} $, $ t\in(0, 1) $, we have
|
$ f(tx+m(1−t)y)≤h(tα)f(x)+mh(1−tα)f(y), $
|
(1.2) |
see [4] for more details.
In [13], authors defined a new notion of functions as follows:
Let $ h: J\subset\mathbb{R}\rightarrow \mathbb{R} $ be a function with $ h\not\equiv0 $. A function $ f:I\rightarrow \mathbb{R} $ is said to be $ (\alpha, h-m) $-convex with respect to a strictly monotone function $ g:K\subset\mathbb{I}\rightarrow \mathbb{R} $, if $ f, h\geq0 $ and for each $ x, y\in I $, we have
|
$ f∘g−1(tx+m(1−t)y)≤h(tα)f∘g−1(x)+mh(1−tα)f∘g−1(y), $
|
(1.3) |
where $ (\alpha, m)\in[0, 1]^{2} $ and $ t\in(0, 1) $. This idea is extended and in [14] refined $ (\alpha, h-m) $-convex function has been introduced which provide refinements of various kinds of convex functions which have been studied in [15,16,17] can be reproduced.
Let $ h:J\subseteq\mathbb{R}\rightarrow \mathbb{R} $ be a function with $ J $ containing $ (0, 1) $. A function $ f:I\rightarrow \mathbb{R} $ is called refined $ (\alpha, h-m) $-convex function, if $ f, h\geq0 $ and for each $ x, y\in I $, we have
|
$ f(tx+m(1−t)y)≤h(tα)h(1−tα)(f(x)+mf(y)), $
|
(1.4) |
where $ (\alpha, m)\in (0, 1]^2 $ and $ t\in(0, 1) $, see [14] for more details.
The aim of this work is to introduce refined $ (\alpha, h-m) $-convex function with respect to strictly monotone function. By applying this new definition we will establish generalized Riemann-Liouville fractional integral inequalities. These inequalities will also provide refinements of Riemann-Liouville fractional integral inequalities for specific functions involved in Definition 2.1.
Fractional calculus operators of differentiation and integration are key factors in generalizing classical concepts of calculus. The role of derivatives and their applications in diverse fields of science and engineering is remarkable. Fractional calculus operators generalize all the classical phenomena related to usual derivative and integration, see [18] for more details.
The Riemann-Liouville fractional and derivative integral operators are the most classical well-known integrals and derivatives of fractional order defined as follows:
Definition 1.1. [18] Let $ f\in L_{1}[u, v] $. Then Riemann-Liouville fractional integrals of order $ \beta $ of a function $ f $ are given as follows:
|
$ Iβu+f(x)=1Γ(β)∫xu(x−t)β−1f(t)dt,x>u, $
|
(1.5) |
|
$ Iβv−f(y)=1Γ(β)∫vy(t−y)β−1f(t)dt,y<v, $
|
(1.6) |
where
| $ \Gamma(\beta) = \int_{0}^{\infty}t^{\beta-1}e^{-t}dt, $ |
and $ \Re(\beta) > 0 $.
Definition 1.2. [19] Let $ f\in L_{1}[u, v] $. Then $ k $-fractional Riemann-Liouville integrals of order $ \beta $ of a function $ f $ are given as follows:
|
$ kIβu+f(x)=1kΓk(β)∫xu(x−t)βk−1f(t)dt,x>u, $
|
(1.7) |
|
$ kIβv−f(y)=1kΓk(β)∫vy(t−y)βk−1f(t)dt,y<v, $
|
(1.8) |
where
| $ \Gamma_k(\beta) = \int_{0}^{\infty}t^{\beta-1}e^{-\frac{t^k}{k}}dt, $ |
and $ \Re(\beta), k > 0 $. Using $ \Gamma_{k}(\beta) = k^{\frac{\beta}{k}-1}\Gamma\left(\frac{\beta}{k}\right) $ in (1.5) and (1.6), one can get
| $ k^{\frac{-\beta}{k}}I_{u^+}^{\beta}f(x) = \,_kI_{u^+}^{\beta}f(x) $ |
and
| $ k^{\frac{-\beta}{k}}I_{v^-}^{\beta}f(y) = \,_kI_{v^-}^{\beta}f(y). $ |
The Hadamard inequality is of immense importance in the theory of inequalities and a lot of work has been published about this inequality in the past few decades, see [20,21,22,23]. The Hadamard inequality is stated as follows:
Let $ f $ be a convex function on $ [u, v] $ with $ u < v $. Then, the following inequality holds:
|
$ f(u+v2)≤1v−u∫vuf(x)dx≤f(u)+f(v)2. $
|
(1.9) |
A number of numerous generalizations and refinements of the inequality (1.9) have been published in recent years, see [11,14,24]. Keeping in view these results we aim to find some new refinements of existing inequalities by using refined $ (\alpha, h-m) $-convex functions with respect to a strictly monotone function. These results are also applicable for $ k $-fractional version of Riemann-Liouville integrals which are also given in this paper.
The rest of the paper is organized as follows: In Section 2, we define a new class of convex functions, namely refined $ (\alpha, h-m) $-convex function with respect to a strictly monotone function and establish Hadamard type inequalities by using this class of functions. The $ k $-analogue versions of these results are given in Section 3. Moreover, the results of this paper are connected with already published inequalities. In Section 4, we give concluding remarks.
Definition 2.1. Let $ h:J\subseteq\mathbb{R}\rightarrow \mathbb{R} $ be a function with $ (0, 1)\subseteq J $. Also, let $ K, I $ are intervals in $ \mathbb{R} $ such that $ K\subset I $. A function $ f:I\rightarrow \mathbb{R} $ is called refined $ (\alpha, h-m) $-convex function with respect to strictly monotone function $ g:K\rightarrow \mathbb{R} $, $ range(g)\subset I $ if $ f, h\geq0 $ and for each $ x, y\in I $, we have
|
$ f∘g−1(tx+m(1−t)y)≤h(tα)h(1−tα)(f∘g−1(x)+mf∘g−1(y)), $
|
(2.1) |
where $ (\alpha, m)\in (0, 1]^2 $ and $ t\in(0, 1) $.
By comparing inequalities (1.3) and (2.1), it can be realized that if $ 0 < h(t) < 1 $, then (2.1) reduces to the following refinement of $ (\alpha, h-m) $-convex function with respect to strictly monotone function $ g $:
|
$ f∘g−1(tx+m(1−t)y)≤h(tα)h(1−tα)(f∘g−1(x)+mf∘g−1(y))=h(tα)h(1−tα)f∘g−1(x)+mh(tα)h(1−tα)f∘g−1(y)≤h(tα)f∘g−1(x)+mh(1−tα)f∘g−1(y). $
|
(2.2) |
Next, we give some examples as follows:
Example 2.1. The function $ x^2+1 $ is $ (1, I_d-1) $-convex as well as refined $ (1, I_d-1) $-convex with respect to $ x $ on $ (0, 1) $. But on $ (1, \infty) $, it is $ (1, I_d-1) $-convex but not refined $ (1, I_d-1) $-convex with respect to $ x $.
Example 2.2. Let $ g(x) = x^p $ for $ x\in(0, \infty) $ and $ p\in \mathbb{R}\setminus\{0\} $. Then $ g $ is strictly monotone function (strictly increasing for $ p > 0 $ and strictly decreasing for $ p < 0 $) and $ g^{-1}(x) = x^{\frac{1}{p}} $. By using Definition 2.1, we have
|
$ f((tx+m(1−t)y))1p)≤h(tα)h(1−tα)(f(x1p)+mf(y1p)). $
|
(2.3) |
By replacing $ x $ with $ x^p $ and $ y $ with $ y^p $, we get
|
$ f((txp+m(1−t)yp))1p)≤h(tα)h(1−tα)(f(x)+mf(y)). $
|
(2.4) |
Inequality (2.4) gives the definition of refined $ (\alpha, h-m)-p $-convex function given in [25]. Whereas for $ p = -1 $, the inequality (2.3) reduces to the following inequality:
|
$ f(1(tx+m(1−t)y))≤h(tα)h(1−tα)(f(1x)+mf(1y)). $
|
(2.5) |
By replacing $ x $ with $ \frac{1}{x} $ and $ y $ with $ \frac{1}{y} $, we get
|
$ f(xy(ty+m(1−t)x))≤h(tα)h(1−tα)(f(x)+mf(y)), $
|
(2.6) |
that is $ f\circ g^{-1} $ is refined harmonically $ (\alpha, h-m) $-convex function defined in [25].
For $ h(t) = t $ in (2.1), we get definition for refined $ (\alpha, m) $-convex function with respect to a strictly monotone function $ g $ stated as follows:
Definition 2.2. A function $ f:I\rightarrow \mathbb{R} $ is called refined $ (\alpha, m) $-convex function with respect to a strictly monotone function $ g:K\subset I\rightarrow \mathbb{R} $, if for each $ x, y\in I $, we have
|
$ f∘g−1(tx+m(1−t)y)≤tα(1−tα)(f∘g−1(x)+mf∘g−1(y)), $
|
(2.7) |
where $ (\alpha, m)\in(0, 1]^{2} $ and $ t\in(0, 1) $.
By taking $ h(t) = t $ and $ m = 1 $, (2.1) gives the definition for refined $ \alpha $-convex function with respect to a strictly monotone function $ g $ stated as follows:
Definition 2.3. A function $ f:I\rightarrow \mathbb{R} $ is called refined $ \alpha $-convex function with respect to a strictly monotone function $ g:K\subset I\rightarrow \mathbb{R} $, if for each $ x, y\in I $, we have
|
$ f∘g−1(tx+(1−t)y)≤tα(1−tα)(f∘g−1(x)+f∘g−1(y)), $
|
(2.8) |
where $ m\in(0, 1] $ and $ t\in(0, 1) $.
By taking $ \alpha = 1 $, (2.1) gives the definition of refined $ (h-m) $-convex function with respect to a strictly monotone function $ g $ stated as follows:
Definition 2.4. Let $ h:J\subseteq\mathbb{R}\rightarrow \mathbb{R} $ be a function with $ J $ containing $ (0, 1) $. A function $ f:I\rightarrow \mathbb{R} $ is called refined $ (h-m) $-convex function with respect to a strictly monotone function $ g:K\subset I\rightarrow \mathbb{R} $, if $ f, h\geq 0 $ and for each $ x, y\in I $, one have the inequality
|
$ f∘g−1(tx+m(1−t)y)≤h(t)h(1−t)(f∘g−1(x)+mf∘g−1(y)), $
|
(2.9) |
where $ m\in[0, 1] $ and $ t\in(0, 1) $.
By taking $ \alpha = 1 = m $, (2.1) gives the definition of refined $ h $-convex function with respect to a strictly monotone function $ g $ stated as follows:
Definition 2.5. Let $ h:J\subseteq\mathbb{R}\rightarrow \mathbb{R} $ be a function with $ J $ containing $ (0, 1) $. A function $ f:I\rightarrow \mathbb{R} $ is called refined $ h $-convex function with respect to a strictly monotone function $ g:K\subset{I}\rightarrow \mathbb{R} $, if $ f, h\geq0 $ and for each $ x, y\in I $, one have the inequality
|
$ f∘g−1(tx+(1−t)y)≤h(t)h(1−t)(f∘g−1(x)+f∘g−1(y)), $
|
(2.10) |
where $ t\in(0, 1) $.
By taking $ \alpha = 1 $ and $ h(t) = t^{s} $, (2.1) gives the definition of refined $ (s, m) $-convex function with respect to a strictly monotone function $ g $ stated as follows:
Definition 2.6. A function $ f:I\rightarrow \mathbb{R} $ is called refined $ (s, m) $-convex function with respect to a strictly monotone function $ g:K\subset I\rightarrow \mathbb{R} $, if for each $ x, y\in I $, we have
|
$ f∘g−1(tx+m(1−t)y)≤ts(1−t)s(f∘g−1(x)+mf∘g−1(y)), $
|
(2.11) |
where $ (s, m)\in(0, 1]^{2} $ and $ t\in(0, 1) $.
By taking $ \alpha = 1 $ and $ h(t) = t $, (2.1) gives the definition of refined $ m $-convex function with respect to a strictly monotone function $ g $ stated as follows:
Definition 2.7. A function $ f:I\rightarrow \mathbb{R} $ is called refined $ m $-convex function with respect to a strictly monotone function $ g:K\subset I\rightarrow \mathbb{R} $, if for each $ x, y\in I $, we have
|
$ f∘g−1(tx+m(1−t)y)≤t(1−t)(f∘g−1(x)+mf∘g−1(y)), $
|
(2.12) |
where $ m\in(0, 1] $ and $ t\in(0, 1) $.
Now, we give the following Hadamard type inequalities for Riemann-Liouville fractional integral operators via refined $ (\alpha, h-m) $-convex function with respect to a strictly monotone function $ g $.
Theorem 2.1. Let $ f:[u, mv] \rightarrow \mathbb{R} $ with $ 0\leq u < mv $ and $ f\in L_{1}[u, v] $. Also, suppose that $ f\geq0 $ is refined $ (\alpha, h-m) $-convex function with respect to a strictly monotone function $ g $. Then for $ (\alpha, m)\in(0, 1]^{2} $, the following fractional integral inequality holds:
| $ \frac{f\circ g^{-1}\left(\frac{u+mv}{2}\right)}{h\left(\frac{1}{2^{\alpha}}\right)h\left(\frac{2^{\alpha}-1}{2^{\alpha}}\right)}\leq\frac{\Gamma({\beta+1})}{(mv-u)^{\beta}}\left[I_{u^+}^{\beta}f\circ g^{-1}(mv)+m^{\beta+1}I_{v^-}^{\beta}f\circ g^{-1}\left(\frac{u}{m}\right)\right]\\ \leq \beta\left[f\circ g^{-1}(u)+2mf\circ g^{-1}(v)+m^2f\circ g^{-1}\left(\frac{u}{m^{2}}\right)\right]\int_{0}^{1}h(t^{\alpha})h(1-t^{\alpha})t^{\beta-1}dt, $ | (2.13) |
with $ \beta > 0 $.
Proof. Since $ f $ is refined $ (\alpha, h-m) $-convex function with respect to a strictly monotone function $ g $, by using (2.1), one can have the following inequality:
|
$ f∘g−1(x+my2)≤h(12α)h(2α−12α)[f∘g−1(x)+mf∘g−1(y)]. $
|
(2.14) |
By setting $ x = tu+m(1-t)v $ and $ y = \frac{u}{m}\left(1-t\right)+vt $ in (2.14), the following inequality can be yielded:
| $ f\circ g^{-1}\left(\frac{u+mv}{2}\right)\\ \leq h\left(\frac{1}{2^{\alpha}}\right)h\left(\frac{2^{\alpha}-1}{2^{\alpha}}\right)\left(f\circ g^{-1}(tu+m(1-t)v)+mf\circ g^{-1}\left(\frac{u}{m}(1-t)+vt\right)\right). $ |
Now, by integrating the above inequality over the interval $ [0, 1] $ after multiplying with $ t^{\beta-1} $, one can have the following inequality:
| $ \frac{f\circ g^{-1}\left(\frac{u+mv}{2}\right)}{\beta h\left(\frac{1}{2^{\alpha}}\right)h\left(\frac{2^{\alpha}-1}{2^{\alpha}}\right)}\\ \leq\int_{0}^{1}f\circ g^{-1}(tu+m(1-t)v)t^{\beta-1}dt+m\int_{0}^{1}f\circ g^{-1}\left(\frac{u}{m}(1-t)+vt\right)t^{\beta-1}dt.$ | (2.15) |
On the right hand side by applying definition of refined $ (\alpha, h-m) $-convexity with respect to a strictly monotone function $ g $, one can obtain the following inequality:
|
(2.16) |
Making change of variables, then by applying Definition 1.1 and multiplying by $ \beta $, one can easily obtain the inequality required in (2.13). The proof is completed.
Remark 2.1. (i) For $ g(x) = x $, (2.13) reduces to [14,Theorem 1].
(ii) For $ m = \alpha = 1 $ and $ h(t) = t $, (2.13) reduces to [24,Theorem 3.1].
(iii) For $ \beta = 1 $ along with the conditions of (ii), (2.13) reduces to [24,Theorem 2.1].
(vi) For $ h(t) = t $ along with the condition of (i), (2.13) reduces to [14,Corollary 1].
(vii) For $ \alpha = 1 $ along with the condition of (i), (2.13) reduces to [14,Corollary 2].
(viii) For $ \alpha = 1 $ and $ h(t) = t^s $ along with the condition of (i), (2.13) reduces to [14,Corollary 4].
(ix) For $ \alpha = 1 $ and $ h(t) = t $ along with the condition of (i), (2.13) reduces to [14,Corollary 5].
(x) For $ m = 1 $ in the result of (x), (2.13) reduces to [14,Corollary 6].
By imposing an additional condition, we get the following result which shows the extension of Theorem 2.1.
Theorem 2.2. Along with} the assumptions of Theorem 2.1, if $ h $ is bounded above by $ \frac{1}{\sqrt{2}} $, then the following inequality holds:
| $ {2}f\circ g^{-1}\left(\frac{u+mv}{2}\right)\leq\frac{1}{h\left(\frac{1}{2^{\alpha}}\right)h\left(\frac{2^{\alpha}-1}{2^{\alpha}}\right)}f\circ g^{-1}\left(\frac{u+mv}{2}\right)\\ \leq\frac{\Gamma({\beta+1})}{(mv-u)^{\beta}}\left[I_{u^+}^{\beta}f\circ g^{-1}(mv)+m^{\beta+1}I_{v^-}^{\beta}f\circ g^{-1}\left(\frac{u}{m}\right)\right]\\ \leq \beta\left[f\circ g^{-1}(u)+2mf\circ g^{-1}(v)+m^2f\circ g^{-1}\left(\frac{u}{m^{2}}\right)\right]\int_{0}^{1}h(t^{\alpha})h(1-t^{\alpha})t^{\beta-1}dt\\ \leq \frac{1}{2}\left[f\circ g^{-1}(u)+2mf\circ g^{-1}(v)+m^2f\circ g^{-1}\left(\frac{u}{m^{2}}\right)\right]. $ | (2.17) |
Proof. By using the condition $ h(t)\leq\frac{1}{\sqrt{2}} $, one can have $ h(t^\alpha)h(1-t^\alpha)\leq\frac{1}{2} $. Integrating over $ [0, 1] $ after multiplying with $ t^{\beta-1} $, we get the following inequality:
|
$ ∫10h(tα)h(1−tα)tβ−1dt≤12β. $
|
(2.18) |
Since $ f $ is refined $ (\alpha, h-m) $-convex function with respect to a strictly monotone function $ g $, by using (2.1), one can have the following inequality:
|
$ 1h(12α)h(2α−12α)f∘g−1(u+mv2)≥2f∘g−1(u+mv2). $
|
(2.19) |
From inequalities (2.13), (2.18) and (2.19), one can get (2.17).
Remark 2.2. (i) For $ g(x) = x $, (2.17) reduces to [14,Theorem 2].
(ii) For $ g(x) = x^p $, (2.17) reduces to [25,Theorem 2].
(iii) For $ \alpha = 1 $ along with the condition of (i), (2.17) reduces to [14,Corollary 3].
(v) For $ \alpha = 1 $ along with the condition of (ii), (2.17) reduces to [25,Corollary 3].
Corollary 2.1. Using $ h(t) = t $, (2.13) gives the fractional integral inequality for refined $ (\alpha, m) $-convex function with respect to a strictly monotone function $ g $ in the following form:
| $ \frac{2^{2\alpha}}{2^{\alpha}-1}f\circ g^{-1}\left(\frac{u+mv}{2}\right)\leq\frac{\Gamma({\beta+1})}{(mv-u)^{\beta}}\left[I_{u^+}^{\beta}f\circ g^{-1}(mv)+m^{\beta+1}I_{v^-}^{\beta}f\circ g^{-1}\left(\frac{u}{m}\right)\right]\\ \leq\frac{\alpha\beta}{(\beta+\alpha)(\beta+2\alpha)}\Big[f\circ g^{-1}(u)+2mf\circ g^{-1}(v)+m^2f\circ g^{-1}\left(\frac{u}{m^{2}}\right)\Big]. $ | (2.20) |
Corollary 2.2. Applying $ \alpha = 1 $, (2.13) gives the fractional integral inequality for refined $ (h-m) $-convex function with respect to strictly monotone function $ g $ in the following form:
| $ \frac{1}{h^{2}\left(\frac{1}{2}\right)}f\circ g^{-1}\left(\frac{u+mv}{2}\right)\leq\frac{\Gamma({\beta+1})}{(mv-u)^{\beta}}\left[I_{u^+}^{\beta}f\circ g^{-1}(mv)+m^{\beta+1}I_{v^-}^{\beta}f\circ g^{-1}\left(\frac{u}{m}\right)\right]\\ \leq\beta\left[f\circ g^{-1}(u)+2mf\circ g^{-1}(v)+m^2f\circ g^{-1}\left(\frac{u}{m^{2}}\right)\right]\int_{0}^{1}h(t)h(1-t)t^{\beta-1}dt. $ | (2.21) |
Corollary 2.3. Applying $ \alpha = 1 $ and $ h(t) = t^{s} $, (2.13) gives the fractional integral inequality for refined $ (s, m) $-convex function with respect to a strictly monotone function $ g $ in the following form:
|
$ 22sf∘g−1(u+mv2)≤Γ(β+1)(mv−u)β[Iβu+f∘g−1(mv)+mβ+1Iβv−f∘g−1(um)]≤β[f∘g−1(u)+2mf∘g−1(v)+m2f∘g−1(um2)]B(1+s,s+β), $
|
(2.22) |
where $ \mathbb{B}(1+s, s+\beta) $ is the beta function.
Corollary 2.4. Applying $ \alpha = 1 $ and $ h(t) = t $, (2.13) gives the inequality for refined $ m $-convex function with respect to a strictly monotone $ g $ in the following form:
|
$ 4f∘g−1(u+mv2)≤Γ(β+1)(mv−u)β[Iβu+f∘g−1(mv)+mβ+1Iβv−f∘g−1(um)]≤β(β+1)(β+2)[f∘g−1(u)+2mf∘g−1(v)+m2f∘g−1(um2)]. $
|
(2.23) |
Corollary 2.5. Applying $ \alpha = 1 $, (2.17) gives the fractional integral inequality for refined $ (h-m) $-convex function with respect to a strictly monotone function $ g $ in the following form:
| $ {2}f\circ g^{-1}\left(\frac{u+mv}{2}\right)\leq\frac{1}{h^{2}\left(\frac{1}{2}\right)}f\circ g^{-1}\left(\frac{u+mv}{2}\right)\\ \leq\frac{\Gamma({\beta+1})}{(mv-u)^{\beta}}\left[I_{u^+}^{\beta}f\circ g^{-1}(mv)+m^{\beta+1}I_{v^-}^{\beta}f\circ g^{-1}\left(\frac{u}{m}\right)\right]\\ \leq \beta\left[f\circ g^{-1}(u)+2mf\circ g^{-1}(v)+m^2f\circ g^{-1}\left(\frac{u}{m^{2}}\right)\right]\int_{0}^{1}h(t)h(1-t)t^{\beta-1}dt\\ \leq \frac{1}{2}\Big[f\circ g^{-1}(u)+2mf\circ g^{-1}(v)+m^2f\circ g^{-1}\left(\frac{u}{m^{2}}\right)\Big]. $ | (2.24) |
Theorem 2.3. Under the assumption of Theorem 2.1, the following fractional integral inequality holds:
|
(2.25) |
with $ \beta > 0 $.
Proof. We use the inequality (2.14) to prove this theorem. By setting $ x = \frac{tu}{2}+m(\frac{2-t}{2})v $ and $ y = \frac{u}{m}(\frac{2-t}{2})+\frac{tv}{2} $ in (2.14), one can get the following inequality:
|
By integrating the above inequality over the interval $ [0, 1] $ after multiplying with $ t^{\beta-1} $, one can have the following inequality:
|
(2.26) |
On the right hand side by applying definition of refined $ (\alpha, h-m) $-convexity with respect to a strictly monotone function $ g $, one can obtain the following inequality:
|
(2.27) |
Making change of variables, then by applying Definition 1.1 and multiplying by $ \beta $, one can easily obtain the inequality required in (2.25).
Remark 2.3. (i) For $ g(x) = x $, (2.25) reduces to [14,Theorem 3].
(ii) For $ \beta = m = \alpha = 1 $ and $ h(t) = t $, (2.25) reduces to [24,Theorem 2.1].
(iii) For $ h(t) = t $ along with the condition of (i), (2.25) reduces to [14,Corollary 7].
(iv) For $ \alpha = 1 $ along with the condition of (i), (2.25) reduces to [14,Corollary 8].
(v) For $ \alpha = 1 $ and $ h(t) = t^s $ along with the condition of (i), (2.25) reduces to [14,Corollary 10].
(vi) For $ \alpha = 1 $ and $ h(t) = t $ along with the condition of (i), (2.25) reduces to [14,Corollary 11].
(vii) For $ m = 1 $ in the result of (i), (2.25) reduces to [14,Corollary 12].
By imposing an additional condition, we get the following result which shows the extension of Theorem 2.3. We will leave the proof for reader.
Theorem 2.4. Along with the assumptions of Theorem 2.3, if $ h $ is bounded above by $ \frac{1}{\sqrt{2}} $, then the following inequality holds:
|
$ 2f∘g−1(u+mv2)≤1h(12α)h(2α−12α)f∘g−1(u+mv2)≤2βΓ(β+1)(mv−u)β[Iβ(u+mv2)+f∘g−1(mv)+mβ+1Iβ(u+mv2m)−f∘g−1(um)]≤β[f∘g−1(u)+2mf∘g−1(v)+m2f∘g−1(um2)]∫10h(tα2α)h(2α−tα2α)tβ−1dt≤12[f∘g−1(u)+2mf∘g−1(v)+m2f∘g−1(um2)]. $
|
(2.28) |
Remark 2.4. (i) For $ g(x) = x $, (2.28) reduces to [14,Theorem 4].
(ii) For $ g(x) = x^p $, (2.28) reduces to [25,Theorem 4].
(iii) For $ \alpha = 1 $ along with the condition of (i), (2.28) reduces to [14,Corollary 9].
(iv) For $ \alpha = 1 $ along with the condition of (ii), (2.28) reduces to [25,Corollary 7].
Corollary 2.6. Applying $ h(t) = t $ in (2.25), the following fractional integral inequality for refined $ (\alpha, m) $-convex function with respect to a strictly monotone function $ g $ is obtained.
|
$ 22α(2α−1)f∘g−1(u+mv2)≤2βΓ(β+1)(mv−u)β[Iβ(u+mv2)+f∘g−1(mv)+mβ+1Iβ(u+mv2m)−f∘g−1(um)]≤β(2α(β+2α)−(β+α))22α(β+α)(β+2α)[f∘g−1(u)+2mf∘g−1(v)+m2f∘g−1(um2)]. $
|
(2.29) |
Corollary 2.7. Applying $ \alpha = 1 $ in (2.25), the following fractional integral inequality for refined $ (h-m) $-convex function with respect to a strictly monotone function $ g $ is obtained.
|
$ 1h2(12)f∘g−1(u+mv2)≤2βΓ(β+1)(mv−u)β[Iβ(u+mv2)+f∘g−1(mv)+mβ+1Iβ(u+mv2m)−f∘g−1(um)]≤β[f∘g−1(u)+2mf∘g−1(v)+m2f∘g−1(um2)]∫10h(t2)h(2−t2)tβ−1dt. $
|
(2.30) |
Corollary 2.8. Applying $ \alpha = 1 $ and $ h(t) = t^{s} $, (2.25) gives the fractional integral inequality for refined $ (s, m) $-convex function with respect to a strictly monotone function $ g $ in the following form:
|
$ 22sf∘g−1(u+mv2)≤2βΓ(β+1)(mv−u)β[Iβ(u+mv2)+f∘g−1(mv)+mβ+1Iβ(u+mv2m)−f∘g−1(um)]≤2β−1β[f∘g−1(u)+2mf∘g−1(v)+m2f∘g−1(um2)]B(s+β,1+s). $
|
(2.31) |
Remark 2.5. If $ m = 1 $ in (2.31), then the result for $ s $-$ tgs $ convex function is obtained.
Corollary 2.9. Applying $ \alpha = 1 $ and $ h(t) = t $, (2.25) gives the inequality for refined $ m $-convex function with respect to a strictly monotone function $ g $ in the following form:
|
$ 4f∘g−1(u+mv2)≤2βΓ(β+1)(mv−u)β[Iβ(u+mv2)+f∘g−1(mv)+mβ+1Iβ(u+mv2m)−f∘g−1(um)]≤β(β+3)4(β+1)(β+2)[f∘g−1(u)+2mf∘g−1(v)+m2f∘g−1(um2)]. $
|
(2.32) |
Corollary 2.10. Applying $ \alpha = 1 $, (2.28) gives the following inequality:
|
$ 2f∘g−1(u+mv2)≤1h2(12)f∘g−1(u+mv2)≤2βΓ(β+1)(mv−u)β[Iβ(u+mv2)+f∘g−1(mv)+mβ+1Iβ(u+mv2m)−f∘g−1(um)]≤β[f∘g−1(u)+2f∘g−1(v)+f∘g−1(um2)]∫10h(t2)h(2−t2)tβ−1dt≤12[f∘g−1(u)+2mf∘g−1(v)+m2f∘g−1(um2)]. $
|
(2.33) |
This section presents extensions of some results of Section 2 into $ k $-fractional versions.
Theorem 3.1. Under the assumption of Theorem 2.1, the following inequality for $ k $-fractional integral holds:
| $ \frac{f\circ g^{-1}\left(\frac{u+mv}{2}\right)}{h\left(\frac{1}{2^{\alpha}}\right)h\left(\frac{2^{\alpha}-1}{2^{\alpha}}\right)}\leq \frac{\Gamma_{k}({\beta+k})}{(mv-u)^{\frac{\beta}{k}}}\left[_{k}I_{u^+}^{\beta}f\circ g^{-1}(mv)+m^{\frac{\beta}{k}+1}_{k}I_{v^-}^{\beta}f\circ g^{-1}\left(\frac{u}{m}\right)\right]\\ \leq \frac{\beta}{k}\left[f\circ g^{-1}(u)+2mf\circ g^{-1}(v)+m^2f\circ g^{-1}\left(\frac{u}{m^{2}}\right)\right]\int_{0}^{1}h(t^{\alpha})h(1-t^{\alpha})t^{\frac{\beta}{k}-1}dt, $ | (3.1) |
with $ \beta, k > 0 $.
Proof. Using (1.7) and (1.8) after replacing $ \beta $ by $ \frac{\beta}{k} $ in the proof of Theorem 2.1, we get the above inequality (3.1).
Theorem 3.2. Under the assumption of Theorem 2.3, for $ k > 0 $, the following inequality holds for $ k $-fractional integrals:
|
$ 1h(12α)h(2α−12α)f∘g−1(u+mv2)≤2βkΓk(β+k)(mv−u)βk[kIβ(u+mv2)+f∘g−1(mv)+mβk+1Iβ(u+mv2m)−f∘g−1(um)]≤βk[f∘g−1(u)+2mf∘g−1(v)+m2f∘g−1(um2)]∫10h(tα2α)h(2α−tα2α)tβk−1dt, $
|
(3.2) |
with $ \beta > 0 $.
Proof. Using (1.7) and (1.8) after replacing $ \beta $ by $ \frac{\beta}{k} $ in the proof of Theorem 2.3, we get the above inequality (3.2).
Remark 3.1. (i) For $ g(x) = x $ all the results of this section coincide with the results of [14,Section 3].
(ii) For $ g(x) = x^p $ all the results of this section coincide with the results of [25,Section 3].
Next, we consider some examples of strictly monotone functions and provide the corresponding inequalities.
Example 3.1. For $ g(x) = x^p $, (2.13) gives the following results for refined $ (\alpha, h-m)-p $-convex function.
Case 3.1. For $ p > 0 $, we have
|
$ f((up+mvp2)1p)h(12α)h(2α−12α)≤Γ(β+1)(mvp−up)β[Iβup+f(mvp)+mβ+1Iβ(mvp)−f(upm)]≤β[f(u)+2mf(v)+m2f(um2)]∫10h(tα)h(1−tα)tβ−1dt. $
|
(3.3) |
Case 3.2. For $ p < 0 $, we have
|
$ f((up+mvp2)1p)h(12α)h(2α−12α)≤Γ(β+1)(up−mvp)β[Iβup+f(mvp)+mβ+1Iβ(mvp)−f(upm)]≤β[f(u)+2mf(v)+m2f(um2)]∫10h(tα)h(1−tα)tβ−1dt. $
|
(3.4) |
Example 3.2. For $ g(x) = \ln(x) $ with $ g^{-1}(x) = e^x $, (2.13) gives the following result:
|
$ f(eu+mv2)h(12α)h(2α−12α)≤Γ(β+1)(mv−u)β[Iβu+f(emv+mβ+1Iβv−f(eum)]≤β[f(eu)+2mf(ev)+m2f(eum2)]∫10h(tα)h(1−tα)tβ−1dt. $
|
(3.5) |
Remark 3.2. (i) For $ h(t) = t $, (3.3) reduces to [25,Corollary 2.1].
(ii) For $ \alpha = 1 $, (3.3) reduces to [25,Corollary 2.2].
(iii) For $ \alpha = 1 $ and $ h(t) = t^s $, (3.3) reduces to [25,Corollary 4].
Example 3.3. For $ g(x) = x^p $, (2.25) gives the following results for refined $ (\alpha, h-m)-p $-convex function.
Case 3.3. For $ p > 0 $, we have
|
(3.6) |
Case 3.4. For $ p < 0 $, we have
|
(3.7) |
Example 3.4. For $ g(x) = \ln(x) $ with $ g^{-1}(x) = e^x $, (2.25) gives the following result:
|
(3.8) |
Remark 3.3. (i) For $ h(t) = t $, (3.6) reduces to [25,Corollary 5].
(ii) For $ \alpha = 1 $, (3.6) reduces to [25,Corollary 6].
(iii) For $ \alpha = 1 $ and $ h(t) = t^s $, (3.6) reduces to [25,Corollary 8].
We have introduced a new class of convex functions that provides refinements of many known classes of functions related with convexity. Inequalities for Riemann-Liouville fractional integral operators have been established for this class of functions. Moreover, connection with already published results is given in the form of remarks and some examples are considered. Furthermore, for $ 0 < h(t) < 1 $, the refinements of the results of [13] can be obtained.
This research received funding support from the National Science, Research and Innovation Fund (NSRF), Thailand.
The authors declare that they have no competing interests.
| [1] |
Reiman EM (2014) Alzheimer's disease and other dementias: advances in 2013. Lancet Neurol 13: 3–5. doi: 10.1016/S1474-4422(13)70257-6
|
| [2] |
Liao L, Cheng D, Wang J, et al. (2004) Proteomic characterization of postmortem amyloid plaques isolated by laser capture microdissection. J Biol Chem 279: 37061–37068. doi: 10.1074/jbc.M403672200
|
| [3] |
Wang Q, Wu J, Rowan MJ, et al. (2005) β‐amyloid inhibition of long‐term potentiation is mediated via tumor necrosis factor. Eur J Neurosci 22: 2827–2832. doi: 10.1111/j.1460-9568.2005.04457.x
|
| [4] | Lecanu L, Papadopoulos V (2013) Modeling Alzheimer's disease with non-transgenic rat models. Alzheimers Res Ther 5: 17. |
| [5] |
Sparks DL, Scheff SW, Hunsaker JC, et al. (1994) Induction of Alzheimer-like β-amyloid immunoreactivity in the brains of rabbits with dietary cholesterol. Exp Neurol 126: 88–94. doi: 10.1006/exnr.1994.1044
|
| [6] |
Refolo LM, Malester B, LaFrancois J, et al. (2000) Hypercholesterolemia accelerates the Alzheimer's amyloid pathology in a transgenic mouse model. Neurobiol Dis 7: 321–331. doi: 10.1006/nbdi.2000.0304
|
| [7] |
Anstey KJ, Lipnicki DM, Low LF (2008) Cholesterol as a risk factor for dementia and cognitive decline: A systematic review of prospective studies with meta-analysis. Am J Geriatr Psychiatry 16: 343–354. doi: 10.1097/01.JGP.0000310778.20870.ae
|
| [8] |
Ansari MA, Scheff SW (2010) Oxidative stress in the progression of Alzheimer disease in the frontal cortex. J Neuropathol Exp Neurol 69: 155–167. doi: 10.1097/NEN.0b013e3181cb5af4
|
| [9] |
McLellan ME, Kajdasz ST, Hyman BT, et al. (2003) In vivo imaging of reactive oxygen species specifically associated with thioflavine S-positive amyloid plaques by multiphoton microscopy. J Neurosci 23: 2212–2217. doi: 10.1523/JNEUROSCI.23-06-02212.2003
|
| [10] |
Li F, Calingasan NY, Yu F, et al. (2004) Increased plaque burden in brains of APP mutant MnSOD heterozygous knockout mice. J Neurochem 89: 1308–1312. doi: 10.1111/j.1471-4159.2004.02455.x
|
| [11] |
Dumont M, Wille E, Stack C, et al. (2009) Reduction of oxidative stress, amyloid deposition, and memory deficit by manganese superoxide dismutase overexpression in a transgenic mouse model of Alzheimer's disease. FASEB J 23: 2459–2466. doi: 10.1096/fj.09-132928
|
| [12] |
Murakami K, Murata N, Noda Y, et al. (2011) SOD1 (copper/zinc superoxide dismutase) deficiency drives amyloid β protein oligomerization and memory loss in mouse model of Alzheimer disease. J Biol Chem 286: 44557–44568. doi: 10.1074/jbc.M111.279208
|
| [13] |
Dröge W (2002) Free radicals in the physiological control of cell function. Physiol Rev 82: 47–95. doi: 10.1152/physrev.00018.2001
|
| [14] |
Tramutola A, Lanzillotta C, Perluigi M, et al. (2017) Butterfield, Oxidative stress, protein modification and Alzheimer disease. Brain Res Bull 133: 88–99. doi: 10.1016/j.brainresbull.2016.06.005
|
| [15] | Mamun AA, Hashimoto M, Katakura M, et al. (2014) neuroprotective effect of madecassoside evaluated using amyloid Β 1-42 -mediated in vitro and in vivo Alzheimer's disease models. Intl J Indigenous Med Plants 47: 1669–1682. |
| [16] |
Olton DS, Samuelson RJ (1976) Remembrance of places passed: spatial memory in rats. J Exp Psychol Anim Behav Process 2: 97–116. doi: 10.1037/0097-7403.2.2.97
|
| [17] |
Jarrard LE, Okaichi H, Steward O, et al. (1984) On the role of hippocampal connections in the performance of place and cue tasks: comparisons with damage to hippocampus. Behav Neurosci 98: 946–954. doi: 10.1037/0735-7044.98.6.946
|
| [18] |
Shevchenko G, Sjödin MO, Malmström D, et al. (2010) Cloud-point extraction and delipidation of porcine brain proteins in combination with bottom-up mass spectrometry approaches for proteome analysis. J Proteome Res 9: 3903–3911. doi: 10.1021/pr100116k
|
| [19] | Stepanichev MY, Zdobnova IM, Zarubenko II, et al. (2007) Studies of the effects of central administration of β-amyloid peptide (25–35): pathomorphological changes in the hippocampus and impairment of spatial memory. Neurosci Behav Physiol 36: 101–106. |
| [20] |
Holscher C, Gengler S, Gault VA, et al. (2007) Soluble beta-amyloid[25-35] reversibly impairs hippocampal synaptic plasticity and spatial learning. Eur J Pharmacol 561: 85–90. doi: 10.1016/j.ejphar.2007.01.040
|
| [21] |
Parihar MS, Brewer GJ (2007) Mitoenergetic failure in Alzheimer disease. Am J Physiol Cell Physiol 292: C8–C23. doi: 10.1152/ajpcell.00232.2006
|
| [22] |
Butterfield DA, Hardas SS, Lange ML (2010) Oxidatively modified glyceraldehyde-3-phosphate dehydrogenase (GAPDH) and Alzheimer's disease: many pathways to neurodegeneration. J Alzheimers Dis 20: 369–393. doi: 10.3233/JAD-2010-1375
|
| [23] |
Minjarez B, Valero Rustarazo ML, Sanchez del Pino MM, et al. (2013) Identification of polypeptides in neurofibrillary tangles and total homogenates of brains with Alzheimer's disease by tandem mass spectrometry. J Alzheimers Dis 34: 239–262. doi: 10.3233/JAD-121480
|
| [24] |
Musunuri S, Wetterhall M, Ingelsson M, et al. (2014) Quantification of the brain proteome in Alzheimer's disease using multiplexed mass spectrometry. J Proteome Res 13: 2056–2068. doi: 10.1021/pr401202d
|
| [25] |
Butterfield DA, Perluigi M, Reed T, et al. (2012) Redox proteomics in selected neurodegenerative disorders: from its infancy to future applications. Antioxid Redox Signal 17: 1610–1655. doi: 10.1089/ars.2011.4109
|
| [26] |
Poon HF, Calabrese V, Calvani M, et al. (2006) Proteomics analyses of specific protein oxidation and protein expression in aged rat brain and its modulation by L-acetylcarnitine: insights into the mechanisms of action of this proposed therapeutic agent for CNS disorders associated with oxidative stress. Antioxid Redox Signal 8: 381–394. doi: 10.1089/ars.2006.8.381
|
| [27] |
Díez‐Vives C, Gay M, García‐Matas S, et al. (2009) Proteomic study of neuron and astrocyte cultures from senescence‐accelerated mouse SAMP8 reveals degenerative changes. J Neurochem 111: 945–955. doi: 10.1111/j.1471-4159.2009.06374.x
|
| [28] |
Blair LJ, Zhang B, Dickey CA (2013) Potential synergy between tau aggregation inhibitors and tau chaperone modulators. Alzheimers Res Ther 5: 41–50. doi: 10.1186/alzrt207
|
| [29] |
Yao J, Irwin RW, Zhao L, et al. (2009) Mitochondrial bioenergetic deficit precedes Alzheimer's pathology in female mouse model of Alzheimer's disease. PNAS 106: 14670–14675. doi: 10.1073/pnas.0903563106
|
| [30] |
Cui Y, Huang M, He Y, et al. (2011) Genetic ablation of apolipoprotein A-IV accelerates Alzheimer's disease pathogenesis in a mouse model. Am J Pathol 178: 1298–1308. doi: 10.1016/j.ajpath.2010.11.057
|
| [31] |
Keeney JTR, Swomley AM, Förster S, et al. (2013) Apolipoprotein A‐I: Insights from redox proteomics for its role in neurodegeneration. Proteomics Clin Appl 7: 109–122. doi: 10.1002/prca.201200087
|
| [32] |
Liu HC, Hu CJ, Chang JG, et al. (2006) Proteomic identification of lower apolipoprotein AI in Alzheimer's disease. Dement Geriatr Cogn Disord 21: 155–161. doi: 10.1159/000090676
|
| [33] |
Corder EH, Saunders AM, Strittmatter WJ, et al. (1993) Gene dose of apolipoprotein E type 4 allele and the risk of Alzheimer's disease in late onset families. Science 261: 921–923. doi: 10.1126/science.8346443
|
| [34] |
Sizova D, Charbaut E, Delalande F, et al. (2007) Proteomic analysis of brain tissue from an Alzheimer's disease mouse model by two-dimensional difference gel electrophoresis. Neurobiol Aging 28: 357–370. doi: 10.1016/j.neurobiolaging.2006.01.011
|
| [35] |
Ellis RJ, Olichney JM, Thal LJ, et al. (1996) Cerebral amyloid angiopathy in the brains of patients with Alzheimer's disease: The CERAD experience, part XV. Neurology 46: 1592–1596. doi: 10.1212/WNL.46.6.1592
|
| [36] |
Kim J, Basak JM, Holtzman DM (2009) The role of apolipoprotein E in Alzheimer's disease. Neuron 63: 287–303. doi: 10.1016/j.neuron.2009.06.026
|
| [37] | Choi J, Gao J, Kim J, et al. (2015) The E3 ubiquitin ligase Idol controls brain LDL receptor expression, ApoE clearance, and Aβ amyloidosis. Sci Transl Med 7: 314ra184. |
| [38] |
Tosto G, Reitz C (2013) Genome-wide association studies in Alzheimer's disease: A review. Curr Neurol Neurosci Rep 13: 381. doi: 10.1007/s11910-013-0381-0
|
| [39] |
Yu JT, Tan L (2012) The role of clusterin in Alzheimer's disease: pathways, pathogenesis and therapy. Mol Neurobiol 45: 314–326. doi: 10.1007/s12035-012-8237-1
|
| [40] |
Hong I, Kang T, Yoo Y, et al. (2013) Quantitative proteomic analysis of the hippocampus in the 5XFAD mouse model at early stages of Alzheimer's disease pathology. J Alzheimers Dis 36: 321–334. doi: 10.3233/JAD-130311
|
| [41] |
Puglielli L, Konopka G, Pack-Chung E, et al. (2001) Acyl-coenzyme A: cholesterol acyltransferase modulates the generation of the amyloid β-peptide. Nat Cell Biol 3: 905–912. doi: 10.1038/ncb1001-905
|
| [42] |
Sjögren M, Mielke M, Gustafson D, et al. (2006) Cholesterol and Alzheimer's disease-is there a relation? Mech Ageing Dev 127: 138–147. doi: 10.1016/j.mad.2005.09.020
|
| [43] |
Cortes-Canteli M, Zamolodchikov D, Ahn HJ, et al. (2012) Fibrinogen and altered hemostasis in Alzheimer's disease. J Alzheimers Dis 32: 599–608. doi: 10.3233/JAD-2012-120820
|
| [44] |
Ahn HJ, Zamolodchikov D, Cortes-Canteli M, et al. (2010) Alzheimer's disease peptide β-amyloid interacts with fibrinogen and induces its oligomerization. Proc Natl Acad Sci U S A 107: 21812–21817. doi: 10.1073/pnas.1010373107
|
| [45] |
Li X, Buxbaum JN (2011) Transthyretin and the brain re-visited: Is neuronal synthesis of transthyretin protective in Alzheimer's disease? Mol Neurodegener 6: 79. doi: 10.1186/1750-1326-6-79
|
| [46] |
Flynn JM, Melov S (2013) SOD2 in mitochondrial dysfunction and neurodegeneration. Free Radic Biol Med 62: 4–12. doi: 10.1016/j.freeradbiomed.2013.05.027
|
| [47] |
De Leo ME, Borrello S, Passantino M, et al. (1998) Oxidative stress and overexpression of manganese superoxide dismutase in patients with Alzheimer's disease. Neurosci Lett 250: 173–176. doi: 10.1016/S0304-3940(98)00469-8
|
| [48] |
Manavalan A, Mishra M, Sze SK, et al. (2013) Brain-site-specific proteome changes induced by neuronal P60TRP expression. Neurosignals 21: 129–149. doi: 10.1159/000343672
|
| [49] |
Sun KH, Chang KH, Clawson S, et al. (2011) Glutathione‐S‐transferase P1 is a critical regulator of Cdk5 kinase activity. J Neurochem 118: 902–914. doi: 10.1111/j.1471-4159.2011.07343.x
|
| [50] |
Luo J, Wärmländer SKTS, Gräslund A, et al. (2014) Non-chaperone proteins can inhibit aggregation and cytotoxicity of Alzheimer amyloid β peptide. J Biol Chem 289: 27766–27775. doi: 10.1074/jbc.M114.574947
|
| [51] |
Habib KL, Lee TCM, Yang J (2010) Inhibitors of catalase-amyloid interactions protect cells from β-amyloid-induced oxidative stress and toxicity. J Biol Chem 285: 38933–38943. doi: 10.1074/jbc.M110.132860
|
| [52] |
Cocciolo A, Di Domenico F, Coccia R, et al. (2012) Decreased expression and increased oxidation of plasma haptoglobin in Alzheimer disease: Insights from redox proteomics. Free Radic Biol Med 53: 1868–1876. doi: 10.1016/j.freeradbiomed.2012.08.596
|
| [53] |
Spagnuolo MS, Maresca B, La Marca V, et al. (2014) Haptoglobin interacts with apolipoprotein E and beta-amyloid and influences their crosstalk. ACS Chem Neurosci 5: 837–847. doi: 10.1021/cn500099f
|
| [54] |
Poynton AR, Hampton MB (2014) Peroxiredoxins as biomarkers of oxidative stress. Biochim Biophys Acta 1840: 906–912. doi: 10.1016/j.bbagen.2013.08.001
|
 neurosci-06-04-299-s001.pdf |
 |
| 1. | Haiying Wang, Dali Hu, 2024, Trapezium type inequality for the n-times differentiable a-preinvex functions, 979-8-3315-4024-1, 78, 10.1109/IHMSC62065.2024.00025 | |
| 2. | Haiying Wang, Zufeng Fu, Xiaolong Fu, Dali Hu, 2024, Hermite-Hadamard type inequality of the n-times differentiable E-preinvex functions, 979-8-3315-4024-1, 74, 10.1109/IHMSC62065.2024.00024 |
Figures(5)
Mohammad Azizur Rahman, Shahdat Hossain, Noorlidah Abdullah, Norhaniza Aminudin. Brain proteomics links oxidative stress with metabolic and cellular stress response proteins in behavioural alteration of Alzheimer’s disease model rats[J]. AIMS Neuroscience, 2019, 6(4): 299-315. doi: 10.3934/Neuroscience.2019.4.299
DownLoad: