Citation: Sergio Branciamore, Andrei S. Rodin, Grigoriy Gogoshin, Arthur D. Riggs. Epigenetics and Evolution: Transposons and the Stochastic Epigenetic Modification Model[J]. AIMS Genetics, 2015, 2(2): 148-162. doi: 10.3934/genet.2015.2.148
| [1] | Zenan Du, Lihua You, Hechao Liu, Yufei Huang . The Sombor index and coindex of two-trees. AIMS Mathematics, 2023, 8(8): 18982-18994. doi: 10.3934/math.2023967 |
| [2] | Chenxu Yang, Meng Ji, Kinkar Chandra Das, Yaping Mao . Extreme graphs on the Sombor indices. AIMS Mathematics, 2022, 7(10): 19126-19146. doi: 10.3934/math.20221050 |
| [3] | Fan Wu, Xinhui An, Baoyindureng Wu . Sombor indices of cacti. AIMS Mathematics, 2023, 8(1): 1550-1565. doi: 10.3934/math.2023078 |
| [4] | Akbar Ali, Sadia Noureen, Akhlaq A. Bhatti, Abeer M. Albalahi . On optimal molecular trees with respect to Sombor indices. AIMS Mathematics, 2023, 8(3): 5369-5390. doi: 10.3934/math.2023270 |
| [5] | Qiaozhi Geng, Shengjie He, Rong-Xia Hao . On the extremal cacti with minimum Sombor index. AIMS Mathematics, 2023, 8(12): 30059-30074. doi: 10.3934/math.20231537 |
| [6] | Juan C. Hernández, José M. Rodríguez, O. Rosario, José M. Sigarreta . Extremal problems on the general Sombor index of a graph. AIMS Mathematics, 2022, 7(5): 8330-8343. doi: 10.3934/math.2022464 |
| [7] | Akbar Ali, Ivan Gutman, Boris Furtula, Abeer M. Albalahi, Amjad E. Hamza . On chemical and mathematical characteristics of generalized degree–based molecular descriptors. AIMS Mathematics, 2025, 10(3): 6788-6804. doi: 10.3934/math.2025311 |
| [8] | Yufei Huang, Hechao Liu . Bounds of modified Sombor index, spectral radius and energy. AIMS Mathematics, 2021, 6(10): 11263-11274. doi: 10.3934/math.2021653 |
| [9] | Xiuwen Wang, Maoqun Wang . Sombor index of uniform hypergraphs. AIMS Mathematics, 2024, 9(11): 30174-30185. doi: 10.3934/math.20241457 |
| [10] | Shabana Anwar, Muhammad Kamran Jamil, Amal S. Alali, Mehwish Zegham, Aisha Javed . Extremal values of the first reformulated Zagreb index for molecular trees with application to octane isomers. AIMS Mathematics, 2024, 9(1): 289-301. doi: 10.3934/math.2024017 |
Ostrowski's Inequality. Let $ f:I\subset [0, +\infty)\to \mathbb{R} $ be a differentiable function on int(I), such that $ f'\in L[a, b] $, where $ a, b\in I $ with $ a < b $. If $ |f'(x)|\le M $ for all $ x\in [a, b] $, then the inequality:
|
$ |f(x)−1b−a∫baf(t)dt|≤M(b−a)[14+(x−a+b2)2(b−a)2], ∀x∈[a,b] $
|
(1.1) |
holds for all $ x\in [a, b] $. This inequality was introduced by Alexander Ostrowski in [26], and with the passing of the years, generalizations on the same, involving derivatives of the function under study, have taken place. It is playing a very important role in all the fields of mathematics, especially in the theory approximations. Thus such inequalities were studied extensively by many researches and numerous generalizations, extensions and variants of them for various kind of functions like bounded variation, synchronous, Lipschitzian, monotonic, absolutely continuous and n-times differentiable mappings etc.
For recent results and generalizations concerning Ostrowski's inequality, we refer the reader to the recent papers [1,3,4,31,32]. The convex functions play a significant role in many fields, for example in biological system, economy, optimization and so on [2,16,19,24,29,34,39]. And many important inequalities are established for these class of functions. Also the evolution of the concept of convexity has had a great impact in the community of investigators. In recent years, for example, generalized concepts such as s-convexity (see[10]), h-convexity (see [30,33]), m-convexity (see [7,15]), MT-convexity (see[21]) and others, as well as combinations of these new concepts have been introduced.
The role of convex sets, convex functions and their generalizations are important in applied mathematics specially in nonlinear programming and optimization theory. For example in economics, convexity plays a fundamental role in equilibrium and duality theory. The convexity of sets and functions have been the object of many studies in recent years. But in many new problems encountered in applied mathematics the notion of convexity is not enough to reach favorite results and hence it is necessary to extend the notion of convexity to the new generalized notions. Recently, several extensions have been considered for the classical convex functions such that some of these new concepts are based on extension of the domain of a convex function (a convex set) to a generalized form and some of them are new definitions that there is no generalization on the domain but on the form of the definition. Some new generalized concepts in this point of view are pseudo-convex functions [22], quasi-convex functions [5], invex functions [17], preinvex functions [25], B-vex functions [20], B-preinvex functions [8], E-convex functions [38], Ostrowski Type inequalities for functions whose derivatives are $ (m, h_1, h_2) $-convex [35], Féjer Type inequalities for $ (s, m) $-convex functions in the second sense [36] and Hermite-Hadamard-Féjer Type inequalities for strongly $ (s, m) $-convex functions with modulus C, in the second sense [9]. In numerical analysis many quadrature rules have been established to approximate the definite integrals. Ostrowski inequality provides the bounds of many numerical quadrature rules [13].
In this paper we have established new Ostrowski's inequality given by Badreddine Meftah in [23] for $ s $-$ \varphi $-convex functions with $ f\in C^n([a, b]) $ such that $ f^{(n)}\in L([a, b]) $ and we give some applications to some special means, the midpoint formula and some examples for the case $ n = 2 $.
Recall that a real-valued function $ f $ defined in a real interval $ J $ is said to be convex if for all $ x, y \in J $ and for any $ t\in [0, 1] $ the inequality
|
$ f(tx+(1−t)y)≤tf(x)+(1−t)f(y) $
|
(2.1) |
holds. If inequality 2.1 is strict when we say that $ f $ is strictly convex, and if inequality 2.1 is reversed the function $ f $ is said to be concave. In [37] we introduced the notion of $ s $-$ \varphi $-convex functions as a generalization of $ s $-convex functions in first sense.
Definition 1. Let $ 0 < s \le 1 $. A function $ f:I\subset \mathbb{R}\to \mathbb{R} $ is called $ s $-$ \varphi $-convex with respect to bifunction $ \varphi:\mathbb{R}\times \mathbb{R}\to \mathbb{R} $ (briefly $ \varphi $-convex), if
|
$ f(tx+(1−t)y)≤f(y)+tsφ(f(x),f(y)) $
|
(2.2) |
for all $ x, y\in I $ and $ t\in [0, 1] $.
Example 1. Let $ f(x) = x^2 $, then $ f $ is convex and $ \frac{1}{2} $-$ \varphi $- convex with $ \varphi(u, v) = 2u+v $, indeed
|
$ f(tx+(1−t)y)=(tx+(1−t)y)2=t2x2+2t(1−t)xy+(1−t)2y2≤y2+tx2+2txy=y2+t12[t12x2+2t12xy]. $
|
On the other hand;
|
$ 0<t<1⟹0<t12<1⟹t12x2+2t12xy≤x2+2xy≤x2+x2+y2. $
|
Hence,
|
$ f(tx+(1−t)y)≤y2+t12[2x2+y2]=f(y)+t12φ(f(x),f(y)). $
|
Example 2. Let $ f(x) = x^n $ and $ 0 < s\le 1 $, then $ f $ is convex and $ s $-$ \varphi $- convex with $ \varphi(u, v) = \sum_{k = 1}^{n}\left(nk\right)v^{1-\frac{k}{n}}\left(u^{\frac{1}{n}}-v^{\frac{1}{n}}\right)^n $, indeed
|
$ f(tx+(1−t)y)=f(y+t(x−y))=(y+t(x−y))n=yn+n∑k=1(nk)yn−k(t(x−y))n=yn+ts[n∑k=1(nk)tn−syn−k(x−y)n]≤yn+ts[n∑k=1(nk)(yn)n−kn((xn)1n−(yn)1n)n]. $
|
Remark 1. If $ f $ is increasing monotone in $ [a, b] $, then $ f $ is $ s $-$ \varphi $- convex for $ \varphi(x, y) = K $, where $ K\in [0, +\infty) $ and $ s\in (0, 1] $.
In this section, we give some integral approximation of $ f\in C^n([a, b]) $ such that $ f^{(n)}\in L([a, b]) $, for $ n\ge 1 $ using the following lemma as the main tool (see [11]).
Lemma 1. Let $ f:[a, b]\to \mathbb{R} $ be a differentiable mapping such that $ f^{(n-1)} $ is absolutely continuous on $ [a, b] $. Then for all $ x\in [a, b] $ we have the identity
| $ \int_a^bf(t)dt = \sum\limits_{k = 0}^{n}\left[\frac{(b-x)^{k+1}+(-1)^k(x-a)^{k+1}}{(k+1)!}\right] f^{(k)}(x)+(-1)^n\int_a^bK_n(x, t)f^{(n)}(t)dt, $ |
where the kernel $ K_n:[a, b]^2\to \mathbb{R} $ is given by
|
$ K_n(x, t) = \left\{ (t−a)nn!ift∈[a,x](t−b)nn!ift∈(x,b] \right. $
|
with $ x\in [a, b] $ and $ n $ is natural number, $ n\ge 1 $.
Theorem 1. Let $ f:I\to \mathbb{R} $ be n-times differentiable on $ [a, b] $ such that $ f^{(n)}\in L([a, b]) $ with $ n\ge 1 $ and $ 0 < s \le 1 $. If $ |f^{(n)}| $ is s-$ \varphi $-convex, then the following inequality
|
$ |∫baf(t)dt−n∑k=0[(b−x)k+1+(−1)k(x−a)k+1(k+1)!]f(k)(x)|≤(x−a)n+1n!(1n+1|f(n)(a)|+1n+s+1φ(|f(n)(a)|,|f(n)(x)|))+(b−x)n+1n![|f(n)(x)|n+1+n∑k=0(nk)(−1)k1k+s+1φ(|f(n)(x)|,|f(n)(b)|)] $
|
holds for all $ x\in [a, b] $.
Proof. From Lemma 1, properties of modulus, making the changes of variables $ u = (1-t)a+tx $ in the first integral and $ u = (1-t)x+tb $ in the second integral we have that,
|
$ |∫baf(t)dt−n∑k=0[(b−x)k+1+(−1)k(x−a)k+1(k+1)!]f(k)(x)|≤∫xa(u−a)nn!|f(n)(u)|du+∫bx(b−u)nn!|f(n)(u)|du=(x−a)n+1n!∫10tn |f(n)((1−t)a+tx)|dt+(b−x)n+1n!∫10(1−t)n |f(n)((1−t)x+tb)|dt. $
|
Since $ |f^{(n)}| $ is s-$ \varphi $- convex (2.2) gives
|
$ |∫baf(t)dt−n∑k=0[(b−x)k+1+(−1)k(x−a)k+1(k+1)!]f(k)(x)|≤(x−a)n+1n!∫10tn(|f(n)(a)|+tsφ(|f(n)(a)|,|f(n)(x)|))dt+(b−x)n+1n!∫10(1−t)n(|f(n)(x)|+tsφ(|f(n)(x)|,|f(n)(b)|))dt=(x−a)n+1n!(1n+1|f(n)(a)|+1n+s+1φ(|f(n)(a)|,|f(n)(x)|))+(b−x)n+1n![|f(n)(x)|n+1+n∑k=0(nk)(−1)k1k+s+1φ(|f(n)(x)|,|f(n)(b)|)] $
|
which is the desired result. The proof is completed.
Remark 2. If we take $ s = 1 $ then obtain a result of Meftah B. (see Theorem 2.1 in [23]).
Corollary 1. Let $ f:I\to \mathbb{R} $ be n-times differentiable on $ [a, b] $ such that $ f^{(n)}\in L([a, b]) $ with $ n\ge 1 $ and $ 0 < s \le 1 $. If $ |f^{(n)}| $ is s-convex in the first sense, we have the following estimate
|
$ |∫baf(t)dt−n∑k=0[(b−x)k+1+(−1)k(x−a)k+1(k+1)!]f(k)(x)|≤s(n+1)!(n+s+1)|f(n)(a)|+(b−x)n+1n!|f(n)(b)|n∑k=0(nk)(−1)kk+s+1+(n+1)[(x−a)n+1(n+s+1)(n+1)!+(b−x)n+1(n+1)!(1n+1−n∑k=0(nk)(−1)kk+s+1)]|f(n)(x)|. $
|
Proof. Taking $ \varphi(u, v) = v-u $ in Theorem 1.
Remark 3. It is important to notice that if $ s = 1 $ we have that $ |f^{(n)}| $ is convex and then obtain the corollary 2.2 of Meftah see [23].
Theorem 2. Let $ f:I\to \mathbb{R} $ be n-times differentiable on $ [a, b] $ such that $ f^{(n)}\in L([a, b]) $ with $ n\ge 1 $, $ 0 < s \le 1 $ and let $ q > 1 $ with $ \frac{1}{p}+\frac{1}{q} = 1 $. If $ |f^{(n)}|^q $ is s-$ \varphi $-convex, then the following inequality holds
|
$ |∫baf(t)dt−n∑k=0[(b−x)k+1+(−1)k(x−a)k+1(k+1)!]f(k)(x)|≤(x−a)n+1(s+1)1q(np+1)1pn!((s+1)|f(n)(a)|q+φ(|f(n)(a)|q,|f(n)(x)|q))1q+(b−x)n+1(s+1)1q(np+1)1pn!((s+1)|f(n)(x)|q+φ(|f(n)(x)|q,|f(n)(b)|q))1q. $
|
Proof. From Lemma 1, properties of modulus, and Holder's inequality, we have
|
$ |∫baf(t)dt−n∑k=0[(b−x)k+1+(−1)k(x−a)k+1(k+1)!]f(k)(x)|≤∫xa(u−a)nn!|f(n)(u)|du+∫bx(b−u)nn!|f(n)(u)|du=(x−a)n+1n!∫10tn |f(n)((1−t)a+tx)|dt+(b−x)n+1n!∫10(1−t)n |f(n)((1−t)x+tb)|dt≤(x−a)n+1n!(∫10tnpdt)1p(∫10|f(n)((1−t)a+tx)|qdt)1q+(b−x)n+1n!(∫10(1−t)npdt)1p(∫10|f(n)((1−t)x+tb)|qdt)1q=(x−a)n+1(np+1)1pn!(∫10|f(n)((1−t)a+tx)|qdt)1q+(b−x)n+1(np+1)1pn!(∫10|f(n)((1−t)x+tb)|qdt)1q. $
|
Since $ |f^{(n)}|^q $ is s-$ \varphi $-convex, we deduce
|
$ |∫baf(t)dt−n∑k=0[(b−x)k+1+(−1)k(x−a)k+1(k+1)!]f(k)(x)|≤(x−a)n+1(np+1)1pn!(∫10(|f(n)(a)|q+tsφ(|f(n)(a)|q,|f(n)(x)|q))dt)1q+(b−x)n+1(np+1)1pn!(∫10(|f(n)(x)|q+tsφ(|f(n)(x)|q,|f(n)(b)|q))dt)1q $
|
|
$ =(x−a)n+1(s+1)1q(np+1)1pn!((s+1)|f(n)(a)|q+φ(|f(n)(a)|q,|f(n)(x)|q))1q+(b−x)n+1(s+1)1q(np+1)1pn!((s+1)|f(n)(x)|q+φ(|f(n)(x)|q,|f(n)(b)|q))1q. $
|
Corollary 2. Let $ f:I\to \mathbb{R} $ be n-times differentiable on $ [a, b] $ such that $ f^{(n)}\in L([a, b]) $ with $ n\ge 1 $, $ 0 < s \le 1 $ and let $ q > 1 $ with $ \frac{1}{p}+\frac{1}{q} = 1 $. If $ |f^{(n)}|^q $ is s-convex in the first sense, then the following inequality holds
|
$ |∫baf(t)dt−n∑k=0[(b−x)k+1+(−1)k(x−a)k+1(k+1)!]f(k)(x)|≤(x−a)n+1(s+1)1q(np+1)1pn!(s|f(n)(a)|q+|f(n)(x)|q)1q+(b−x)n+1(s+1)1q(np+1)1pn!(s|f(n)(x)|q+|f(n)(b)|q)1q. $
|
(3.1) |
Proof. Taking $ \varphi(u, v) = v-u $ in Theorem 1.
Corollary 3. Let $ f:I\to \mathbb{R} $ be n-times differentiable on $ [a, b] $ such that $ f^{(n)}\in L([a, b]) $ with $ n\ge 1 $, $ 0 < s \le 1 $ and let $ q > 1 $ with $ \frac{1}{p}+\frac{1}{q} = 1 $. If $ |f^{(n)}|^q $ is s-convex in the first sense, then the following inequality holds
|
$ |∫baf(t)dt−n∑k=0[(b−x)k+1+(−1)k(x−a)k+1(k+1)!]f(k)(x)|≤(x−a)n+1(s+1)1q(np+1)1pn!(s|f(n)(a)|+|f(n)(x)|)+(b−x)n+1(s+1)1q(np+1)1pn!(s|f(n)(x)|+|f(n)(b)|). $
|
Proof. Taking $ \varphi(u, v) = v-u $ in Theorem 1, we obtain 3.1. Then using the following algebraic inequality for all $ a, b\ge 0 $, and $ 0\le \alpha \le 1 $ we have $ (a+b)^{\alpha}\le a^{\alpha}+ b^{\alpha} $, we get the desired result.
Theorem 3. Let $ q > 1 $ and $ f:I\to \mathbb{R} $ be n-times differentiable on $ [a, b] $ such that $ f^{(n)}\in L([a, b]) $ with $ n\ge 1 $, $ 0 < s \le 1 $. If $ |f^{(n)}|^q $ is s-$ \varphi- $convex, then the following inequality
|
$ |∫baf(t)dt−n∑k=0[(b−x)k+1+(−1)k(x−a)k+1(k+1)!]f(k)(x)|≤(n+1)1q(x−a)n+1(n+1)!(1n+1|f(n)(a)|q+1n+s+1φ(|f(n)(a)|q,|f(n)(x)|q))1q+(n+1)1q(b−x)n+1(n+1)!(1n+1|f(n)(x)|q+φ(|f(n)(x)|q,|f(n)(b)|q)n∑k=0(nk)(−1)kk+s+1)1q $
|
holds for all $ x\in [a, b] $.
Proof. From Lemma 1, properties of modulus, and power mean inequality, we have
|
$ |∫baf(t)dt−n∑k=0[(b−x)k+1+(−1)k(x−a)k+1(k+1)!]f(k)(x)|≤∫xa(u−a)nn!|f(n)(u)|du+∫bx(b−u)nn!|f(n)(u)|du=(x−a)n+1n!∫10tn |f(n)((1−t)a+tx)|dt+(b−x)n+1n!∫10(1−t)n |f(n)((1−t)x+tb)|dt $
|
|
$ ≤(x−a)n+1n!(∫10tndt)1−1q(∫10tn |f(n)((1−t)a+tx)|qdt)1q+(b−x)n+1n!(∫10(1−t)ndt)1−1q(∫10(1−t)n |f(n)((1−t)x+tb)|qdt)1q=(n+1)1q(x−a)n+1(n+1)!(∫10tn |f(n)((1−t)a+tx)|qdt)1q+(n+1)1q(b−x)n+1(n+1)!(∫10(1−t)n |f(n)((1−t)x+tb)|qdt)1q. $
|
Since $ |f^{(n)}|^q $ is s-$ \varphi $-convex, we deduce
|
$ |∫baf(t)dt−n∑k=0[(b−x)k+1+(−1)k(x−a)k+1(k+1)!]f(k)(x)|≤(n+1)1q(x−a)n+1(n+1)!(|f(n)(a)|q∫10tndt+φ(|f(n)(a)|q,|f(n)(x)|q)∫10tn+sdt)1q+(n+1)1q(b−x)n+1(n+1)!(|f(n)(x)|q∫10(1−t)ndt+φ(|f(n)(x)|q,|f(n)(b)|q)∫10ts(1−t)ndt)1q=(n+1)1q(x−a)n+1(n+1)!(1n+1|f(n)(a)|q+1n+s+1φ(|f(n)(a)|q,|f(n)(x)|q))1q+(n+1)1q(b−x)n+1(n+1)!(1n+1|f(n)(x)|q+φ(|f(n)(x)|q,|f(n)(b)|q)n∑k=0(nk)(−1)kk+s+1)1q. $
|
The proof is completed.
Remark 4. If we take $ s = 1 $ then obtain a result of Meftah B. (see Theorem 2.6 in [23]).
Corollary 4. Let $ f:I\to \mathbb{R} $ be n-times differentiable on $ [a, b] $ such that $ f^{(n)}\in L([a, b]) $ with $ n\ge 1 $, $ 0 < s \le 1 $ and let $ q > 1 $. If $ |f^{(n)}|^q $ is $ s $-convex in the first sense, then the following inequality
|
$ |∫baf(t)dt−n∑k=0[(b−x)k+1+(−1)k(x−a)k+1(k+1)!]f(k)(x)|≤(n+1)1q(x−a)n+1(n+s+1)1q(n+1)!(s|f(n)(a)|qn+1+|f(n)(x)|q)1q+(n+1)1q(b−x)n+1(n+1)!(1n+1|f(n)(x)|q+[|f(n)(b)|q−|f(n)(x)|q]n∑k=0(nk)(−1)kk+s+1)1q $
|
holds for all $ x\in [a, b] $.
Proof. Taking $ \varphi(u, v) = v-u $ in Theorem 3.
Theorem 4. Let $ f:I\to \mathbb{R} $ be n-times differentiable on $ [a, b] $ such that $ f^{(n)}\in L([a, b]) $ with $ n\ge 1 $, $ 0 < s \le 1 $ and let $ q > 1 $. If $ |f^{(n)}|^q $ is s-$ \varphi $-convex, then the following inequality
|
$ |∫baf(t)dt−n∑k=0[(b−x)k+1+(−1)k(x−a)k+1(k+1)!]f(k)(x)|≤(x−a)n+1n!(1qn+1|f(n)(a)|q+1qn+s+1φ(|f(n)(a)|q,|f(n)(x)|q))1q+(b−x)n+1n!(1qn+1|f(n)(x)|q+qn∑k=0(qnk)(−1)kk+s+1φ(|f(n)(x)|q,|f(n)(b)|q))1q $
|
holds for all $ x\in [a, b] $.
Proof. From Lemma 1, properties of modulus, and power mean inequality, we have
|
$ |∫baf(t)dt−n∑k=0[(b−x)k+1+(−1)k(x−a)k+1(k+1)!]f(k)(x)|≤∫xa(u−a)nn!|f(n)(u)|du+∫bx(b−u)nn!|f(n)(u)|du=(x−a)n+1n!∫10tn |f(n)((1−t)a+tx)|dt+(b−x)n+1n!∫10(1−t)n |f(n)((1−t)x+tb)|dt≤(x−a)n+1n!(∫10dt)1−1q(∫10tqn |f(n)((1−t)a+tx)|qdt)1q+(b−x)n+1n!(∫10dt)1−1q(∫10(1−t)qn |f(n)((1−t)x+tb)|qdt)1q=(x−a)n+1n!(∫10tqn |f(n)((1−t)a+tx)|qdt)1q+(b−x)n+1n!(∫10(1−t)qn |f(n)((1−t)x+tb)|qdt)1q. $
|
Since $ |f^{(n)}|^q $ is s-$ \varphi $-convex, we deduce
|
$ |∫baf(t)dt−n∑k=0[(b−x)k+1+(−1)k(x−a)k+1(k+1)!]f(k)(x)|≤(x−a)n+1n!(|f(n)(a)|q∫10tqndt+φ(|f(n)(a)|q,|f(n)(x)|q)∫10tqn+sdt)1q+(b−x)n+1n!(|f(n)(x)|q∫10(1−t)qndt+φ(|f(n)(x)|q,|f(n)(b)|q)∫10ts(1−t)qndt)1q $
|
|
$ =(x−a)n+1n!(1qn+1|f(n)(a)|q+1qn+s+1φ(|f(n)(a)|q,|f(n)(x)|q))1q+(b−x)n+1n!(1qn+1|f(n)(x)|q+qn∑k=0(qnk)(−1)kk+s+1φ(|f(n)(x)|q,|f(n)(b)|q))1q $
|
which in the desired result.
Remark 5. If we take $ s = 1 $ then obtain a result of Meftah B. (see Theorem 2.9 in [23]).
Corollary 5. Let $ f:I\to \mathbb{R} $ be n-times differentiable on $ [a, b] $ such that $ f^{(n)}\in L([a, b]) $ with $ n\ge 1 $, $ 0 < s \le 1 $ and let $ q > 1 $. If $ |f^{(n)}|^q $ is $ s $-convex in the first sense, then the following inequality
|
$ |∫baf(t)dt−n∑k=0[(b−x)k+1+(−1)k(x−a)k+1(k+1)!]f(k)(x)|≤(x−a)n+1n!(1qn+1|f(n)(a)|q+|f(n)(x)|q−|f(n)(a)|qqn+s+1)1q+(b−x)n+1n!(1qn+1|f(n)(x)|q+qn∑k=0(qnk)(−1)kk+s+1(|f(n)(b)|q−|f(n)(x)|q))1q $
|
holds for all $ x\in [a, b] $.
Proof. Taking $ \varphi(u, v) = v-u $ in Theorem 4.
In this section, using [12] we define $ s $-$ \varphi_b $-convex function as generalized form of $ s $-$ \varphi $ convex functions [37] and give some results.
Definition 2. Let $ \mathbb{R}^+ $ be the set of nonnegative real numbers and $ b:\mathbb{R}\times \mathbb{R}\times [0, 1]\to \mathbb{R}^+ $ be a function with $ t^sb(x, y, t)\in [0, 1] $ for all $ x, y \in \mathbb{R} $, $ t \in [0, 1] $ and $ s \in (0, 1] $. A function $ f:I\to \mathbb{R} $ is called $ s $-$ \varphi_b $-convex if
|
$ f(tx+(1−t)y)≤f(y)+tsb(x,y,t)φ(f(x),f(y)) $
|
for all $ x, y\in \mathbb{R} $ and $ t\in [0, 1] $.
Remark 6. If $ b(x, y, z) = 1 $ then the definition of $ s $-$ \varphi_b $-convex function matches the definition of $ s $-$ \varphi $-convex function.
Theorem 5. Consider a function $ f:I\to \mathbb{R} $ and $ b:\mathbb{R}\times \mathbb{R}\times [0, 1]\to \mathbb{R}^+ $ be a function with $ t^sb(x, y, t)\in [0, 1] $ for all $ x, y \in \mathbb{R} $ and $ s, t \in [0, 1] $. Then the following assertions are equivalent:
(i) $ f $ is $ s $-$ \varphi_b $-convex for some $ b $ and $ s\in [0, 1] $.
(ii) $ f $ is $ \varphi $-quasiconvex.
Proof. $ (i)\to (ii) $ For any $ x, y \in I $ and $ t\in [0, 1] $,
|
$ f(tx+(1−t)y)≤f(y)+tsb(x,y,t)φ(f(x),f(y))≤max{f(y),f(y)+φ(f(x),f(y))}. $
|
$ (ii)\to (i) $ For $ x, y\in I $ and $ t\in [0, 1] $, define
|
$ b(x, y, t) = \left\{ 1ts if t∈[0,1] and f(y)≤f(y)+φ(f(x),f(y))0 if t=0 or f(y)>f(y)+φ(f(x),f(y)) \right. $
|
Notice that $ t^sb(x, y, t)\in [0, 1] $. For a such function $ b $ we have
|
$ f(tx+(1−t)y)≤max{f(y),f(y)+φ(f(x),f(y))}=tsb(x,y,t)[f(y)+φ(f(x),f(y))+(1−tsb(x,y,t))]f(y)=f(y)+tsb(x,y,t)φ(f(x),f(y)). $
|
Remark 7. Let $ f:I\to \mathbb{R} $ be a $ s $-$ \varphi $-convex function. For $ x_1, x_2 \in I $ and $ \alpha_1+\alpha_2 = 1 $, we have $ f(\alpha_1x_1+\alpha_2x_2)\le f(x_2)+\alpha_1^s\varphi(f(x_1), f(x_2)) $. Aso when $ n > 2 $, for $ x_1, x_2, ..., x_n\in I $, $ \sum_{i = 1}^{n}\alpha_i = 1 $ and $ T_i = \sum_{j = 1}^{i}\alpha_j $, we have
|
$ f(n∑i=1αixi)=f((Tn−1n−1∑i=1αiTn−1xi)+αnxn)≤f(xn)+Tsn−1φ(f(n−1∑i=1αiTn−1xi),f(xn)). $
|
(4.1) |
Theorem 6. Let $ f:I\to \mathbb{R} $ be a $ s $-$ \varphi $-convex function and $ \varphi $ be nondecreasing nonnegatively sublinear in first variable. If $ T_i = \sum_{j = 1}^{i}\alpha_j $ for $ i = 1, 2, ..., n $ such that $ T_n = 1 $, then
| $ f\Big(\sum\limits_{i = 1}^{n}\alpha_ix_i\Big)\le f(x_n)+\sum\limits_{i = 1}^{n-1}T_i^s\varphi_f(x_i, x_{i+1}, ..., x_n), $ |
where $ \varphi_f(x_i, x_{i+1}, ..., x_n) = \varphi(\varphi_f(x_i, x_{i+1}, ..., x_{n-1}), f(x_n)) $ and $ \varphi_f(x) = f(x) $ for all $ x\in I $.
Proof. Since $ \varphi $ is nondecreasing nonnegatively sublinear on first variable, so from (4.1) it follows that:
|
$ f(n∑i=1αixi)=f((Tn−1n−1∑i=1αiTn−1xi)+αnxn)≤f(xn)+Tsn−1φ(f(n−1∑i=1αiTn−1xi),f(xn))=f(xn)+(Tn−1)sφ(f(Tn−2Tn−1n−2∑i=1αiTn−2xi+αn−1Tn−1xn−1),f(xn))≤f(xn)+(Tn−1)sφ(f(xn−1)+(Tn−2Tn−1)sφ(f(n−2∑i=1αiTn−2xi),f(xn−1)),f(xn))≤f(xn)+(Tn−1)sφ(f(xn−1),f(xn))+(Tn−2)sφ(φ(f(n−2∑i=1αiTn−2xi),f(xn−1)),f(xn))≤... $
|
|
$ ≤f(xn)+(Tn−1)sφ(f(xn−1),f(xn))+(Tn−2)sφ(φ(f(xn−2),f(xn−1)),f(xn))+...+Ts1φ(φ(...φ(φ(f(x1),f(x2)),f(x3)...),f(xn−1)),f(xn))=f(xn)+(Tn−1)sφf(xn−1,xn)+(Tn−2)sφf(xn−2,xn−1,xn)+...+(T1)sφf(x1,x2,...,xn−1,xn)=f(xn)+n−1∑i=1Tsiφf(xi,xi+1,...,xn). $
|
Example 3. Consider $ f(x) = x^2 $ and $ \varphi(x, y) = 2x+y $ for $ x, y\in \mathbb{R}^+ = [0, +\infty) $. The function $ \varphi $ is nondecreasing nonnegatively sublinear in first variable and $ f $ is $ \frac{1}{2} $-$ \varphi $-convex (see Example 1). Now for $ x_1, x_2, ..., x_n\in \mathbb{R}^+ $ and $ \alpha_1, \alpha_2, ..., \alpha_n $ with $ \sum_{i = 1}^{n}\alpha_i = 1 $ according to Theorem 6 we have
|
$ (n∑i=1αixi)2≤(xn)2+n−1∑i=1T12iφf(xi,xi+1,...,xn)≤(xn)2+n−1∑i=1T12i[2[...2[2x2i+x2i+1]+x2i+2]+...+x2n]. $
|
In this section we give some applications for the special case where $ n = 2 $ and the function $ \varphi(f(x), f(y)) = f(y)-f(x) $, in this case we have that $ f $ is $ s $-convex in the first sense.
Example 4. Let $ s\in (0, 1) $ and $ p, q, r \in \mathbb{R} $, we define the function $ f:[0, +\infty)\to \mathbb{R} $ as
|
$ f(t) = \left\{ pift=0qts+rift>0 \right. , $
|
we have that if $ q\ge 0 $ and $ r\le p $, then $ f $ is s-convex in the first sense (see [18]). If we do $ \varphi(f(x), f(y)) = f(x)-f(y) $, then $ f $ is s-$ \varphi $-convex, but is not $ \varphi $-convex because $ f $ is not convex.
Example 5. In the previous example if $ s = \frac{1}{2} $, $ p = 1 $, $ q = 2 $ and $ r = 1 $ we have that $ f:[0, +\infty)\to \mathbb{R}, f(t) = 2t^{\frac{1}{2}}+1 $ is $ \frac{1}{2} $-$ \varphi $-convex. Then if we define $ g:[0, +\infty)\to \mathbb{R} $, $ g(t) = \frac{8}{15}t^{\frac{5}{2}}+\frac{t^2}{2} $, we have to $ g''(t) = 2t^{\frac{1}{2}}+1 $ is $ \frac{1}{2} $-$ \varphi $-convex in $ [0, +\infty) $ with $ \varphi(f(x), f(y)) = f(x)-f(y) $. Using Theorem 1, for $ a, b\in [0, +\infty) $ with $ a < b $ and $ x\in [a, b] $, we get
|
$ |16(b72−a72)+35(a3−b3)√x+35(b2−a2)x32+21(a−b)x52|≤352(1+2√b)(b−x)3−103(72+√a+6√b)(a−x)3. $
|
Remark 6. In particular if we choose $ a = 0 $ and $ b = 1 $, we have for $ x\in [0, 1] $, we get a graphic representation of the Example 5.
|
Example 6. If we define $ g(t) = \frac{t^4}{12} $ we have that $ g''(t) $ is $ \frac{1}{2} $-$ \varphi $- convex with $ \varphi(u, v) = 2u+v $ (see example 1) and by Theorem 1, for $ a, b\in \mathbb{R} $ with $ a < b $ and $ x\in [a, b] $, we have that
|
$ |b5−a560−(b−a)12x4−[b2−2x(b−a)−a23]x3−[(b−x)3+(x−a)36]x2|≤(x−a)3[x2−56a2]+(b−x)3[19210x2+8105b2]. $
|
Moreover, if choose $ x = \frac{a+b}{2} $, we obtain that
|
$ |b5−a560−(b−a)(a+b)4192−(b−a)3(a+b)296|≤(b−a)316[a23+9a2+2ab+b214+(a+b)212+8a2+16ab+24b2105]. $
|
Then
|
$ |(a−b)5|≤(b−a)37(477a2+194ab+161b2). $
|
Therefore
|
$ (a−b)2≤477a2+194ab+161b27. $
|
Example 7. If we define $ g(t) = \frac{36}{91}\sqrt[3]{2} \ t^{\frac{13}{6}} $ we have that $ |g''(t)|^3 $ is $ \frac{1}{2} $-$ \varphi $-convex with $ \varphi(u, v) = 2u+v $ (see example 1) and by Theorem 4, for $ a, b\in \mathbb{R} $ with $ a < b $ and $ x\in [a, b] $, we have
|
$ |2161729[b196−a196]−36x13691(b−a)+6x7614[(b−x)2−(x−a)2]+x166[(b−x)3+(x−a)3]|≤(x−a)323√48(a12+2x12)13+(b−x)323√48(x12+2b12)13. $
|
In this paper we have established new Ostrowski's inequality given by Badreddine Meftah in [23] for $ s-\varphi- $convex functions with $ f\in C^n([a, b]) $ such that $ f^{(n)}\in L([a, b]) $ with $ n\ge 1 $ and we give some applications to some special means, the midpoint formula and some examples for the case $ n = 2 $. We expect that the ideas and techniques used in this paper may inspire interested readers to explore some new applications of these newly introduced explore some new applications of these newly introduced functions in various fields of pure and applied sciences.
The authors want to give thanks to the Dirección de investigación from Pontificia Universidad Católica del Ecuador for technical support to our research project entitled: "Algunas desigualdades integrales para funciones convexas generalizadas y aplicaciones".
The authors declare that they have no conflicts of interest.
| [1] | M. B. Ekram, K. Kang, H. Kim and J. Kim Retrotransposons as a major source of epigenetic variations in the mammalian genome. Epigenetics : official journal of the DNA Methylation Society, 7 (2012), 370-382. |
| [2] | T. H. Bestor, J. R. Edwards and M. Boulard Notes on the role of dynamic DNA methylation in mammalian development. Proceedings of the National Academy of Sciences of the United States of America,(2014), in press. |
| [3] | M. Thomas, L. Pingault, A. Poulet, J. Duarte, M. Throude, S. Faure, J. P. Pichon, E. Paux, A. V. Probst and C. Tatout Evolutionary history of Methyltransferase 1 genes in hexaploid wheat. BMC genomics, 15 (2014), 922. |
| [4] | H. L. Levin and J. V. Moran Dynamic interactions between transposable elements and their hosts. Nature reviews. Genetics,12 (2011), 615-627. |
| [5] | C. Feschotte Transposable elements and the evolution of regulatory networks. Nature reviews. Genetics,9 (2008),397-39405. |
| [6] | E. Heard and R. A. Martienssen Transgenerational epigenetic inheritance: myths and mechanisms. Cell,157 (2014),95-95109. |
| [7] | C. F. Kratochwil and A. Meyer Closing the genotype-phenotype gap: emerging technologies for evolutionary genetics in ecological model vertebrate systems. BioEssays : news and reviews in molecular, cellular and developmental biology,37 (2015), 213-226. |
| [8] | A. P. Feinberg and R. A. Irizarry Evolution in health and medicine Sackler colloquium: Stochastic epigenetic variation as a driving force of development, evolutionary adaptation, and disease. Proceedings of the National Academy of Sciences of the United States of America, 107 Suppl 1 (2010), 1757-64. |
| [9] | E. Jablonka Epigenetic variations in heredity and evolution. Clinical pharmacology and therapeutics, 92 (2012),683-8. |
| [10] | T. E. Keller and S. V. Yi DNA methylation and evolution of duplicate genes. Proceedings of the National Academy of Sciences of the United States of America, 111 (2014), 5932-7. |
| [11] | M. M. Darby and S. Sabunciyan Repetitive elements and epigenetic marks in behavior and psychiatric disease. Advances in genetics, 86 (2014), 185-252. |
| [12] | S. Branciamore, A. S. Rodin, A. D. Riggs and S. N. Rodin Enhanced evolution by stochastically variable modification of epigenetic marks in the early embryo. Proceedings of the National Academy of Sciences of the United States of America, 111 (2014), 6353-8. |
| [13] | S. N. Rodin and A. D. Riggs Epigenetic silencing may aid evolution by gene duplication. Journal of molecular evolution, 56 (2003), 718-29. |
| [14] | S. N. Rodin, D. V. Parkhomchuk, A. S. Rodin, G. P. Holmquist and A. D. Riggs Repositioning-dependent fate of duplicate genes. DNA and cell biology, 24 (2005), 529-42. |
| [15] | M. R. Reed, A. D. Riggs and J. R. Mann Deletion of a direct repeat element has no effect on Igf2 and H19 imprinting. Mammalian genome, 12 (2001), 873-6. |
| [16] | A. Chess Mechanisms and consequences of widespread random monoallelic expression. Nature reviews. Genetics, 13 (2012), 421-8. |
| [17] | A. M. Cotton, E. M. Price, M. J. Jones, B. P. Balaton, M. S. Kobor and C. J. Brown Landscape of DNA methylation on the X chromosome reffects CpG density, functional chromatin state and X-chromosome inactivation. Human molecular genetics,(2014), in press. |
| [18] | A. V. Gendrel, M. Attia, C. J. Chen, P. Diabangouaya, N. Servant, E. Barillot and E. Heard Developmental dynamics and disease potential of random monoallelic gene expression. Developmental cell, 28 (2014), 366-80. |
| [19] | A. A. Adegbola, G. F. Cox, E. M. Bradshaw, D. A. Ha er, A. Gimelbrant and A. Chess Monoallelic expression of the human FOXP2 speech gene. Proceedings of the National Academy of Sciences of the United States of America,(2014), in press. |
| [20] | M. A. Eckersley-Maslin, D. Thybert, J. H. Bergmann, J. C. Marioni, P. Flicek and D. L. Spector Random monoallelic gene expression increases upon embryonic stem cell differentiation. Developmental cell, 28 (2014), 351-65. |
| [21] | J. P. Shonkoff et. al. Early Experiences Can Alter Gene Expression and Affect Long-Term Development: Working Paper No. 10. National Scientific Council on the Developing Child (2010). |
| [22] | T. Tollefsbol Handbook of Epigenetics, The New Molecular and Medical Genetics 1st edition, Academic Press San Diego, 2011 |
| [23] | C. Weinhouse, O. S. Anderson, T. R. Jones, J. Kim, S. A. Liberman, M. S. Nahar, L. S. Rozek, R. L. Jirtle and D. C. Dolinoy An expression microarray approach for the identification of metastable epialleles in the mouse genome. Epigenetics : official journal of the DNA Methylation Society, 6 (2011), 1105-13. |
| [24] | D. C. Dolinoy, C. Weinhouse, T. R. Jones, L. S. Rozek and R. L. Jirtle Variable histone modifications at the A(vy) metastable epiallele. Epigenetics, 5 (2010), 637-44. |
| [25] | M. Francescatto, M. Vitezic, P. Heutink and A. Saxena Brain-specific noncoding RNAs are likely to originate in repeats and may play a role in up-regulating genes in cis. The international journal of biochemistry & cell biology,54 (2014), 331-7. |
| [26] | E. Maclary, E. Buttigieg, M. Hinten, S. Gayen, C. Harris, M. K. Sarkar, S. Purushothaman and S. Kalantry Differentiation-dependent requirement of Tsix long non-coding RNA in imprinted X-chromosome inactivation. Nature communications, 5 (2014), 4209. |
| [27] | J. H. Thomas and S. Schneider Coevolution of retroelements and tandem zinc finger genes. Genome research, 21 (2011), 1800-12. |
| [28] | Z. D. Smith, M. M. Chan, T. S. Mikkelsen, H. Gu, A. Gnirke, A. Regev and A. Meissner A unique regulatory phase of DNA methylation in the early mammalian embryo. Nature, 484 (2012), 339-44. |
| [29] | H. Guo, P. Zhu, L. Yan, R. Li, B. Hu, Y. Lian, J. Yan, X. Ren, S. Lin, J. Li, X. Jin, X. Shi, P. Liu, X. Wang, W. Wang, Y. Wei, X. Li, F. Guo, X. Wu, X. Fan, J. Yong, L. Wen, S. X. Xie, F. Tang and J. Qiao The DNA methylation landscape of human early embryos. Nature, 511 (2014), 606-10. |
| [30] | Z. D. Smith, M. M. Chan, K. C. Humm, R. Karnik, S. Mekhoubad, A. Regev, K. Eggan and A. Meissner DNA methylation dynamics of the human preimplantation embryo. Nature, 511 (2014), 611-5. |
| [31] | Z. X. Chen and A. D. Riggs DNA methylation and demethylation in mammals. The Journal of biological chemistry,286 (2011), 18347-53. |
| [32] | S. B. Baylin and P. A. Jones A decade of exploring the cancer epigenome - biological and translational implications. Nature reviews. Cancer, 11 (2011), 726-34. |
| [33] | Z. D. Smith and A. Meissner DNA methylation: roles in mammalian development. Nature reviews. Genetics,14 (2013), 204-220. |
| [34] | S. M. Cullen, A. Mayle, L. Rossi and M. A. Goodell Hematopoietic stem cell development: an epigenetic journey. Current topics in developmental biology, 107 (2014), 39-75. |
| [35] | M. Harris High-frequency induction by 5-azacytidine of proline independence in CHO-K1 cells. Somatic cell and molecular genetics, 10 (1984), 615-24. |
| [36] | Y. Shen, F. Yue, D. F. McCleary, Z. Ye, L. Edsall, S. Kuan, U. Wagner, J. Dixon, L. Lee, V. V. Lobanenkov and B. Ren A map of the cis-regulatory sequences in the mouse genome. Nature, 488 (2012), 116-20. |
| [37] | S. C. Elgin and G. Reuter Position-effect variegation, heterochromatin formation, and gene silencing in Drosophila. Cold Spring Harbor perspectives in biology, 5 (2013), a017780. |
| [38] | L. H. Chadwick, L. M. Pertz, K. W. Broman, M. S. Bartolomei and H. F. Willard Genetic control of X chromosome inactivation in mice: definition of the Xce candidate interval. Genetics, 173 (2006), 2103-10. |
| [39] | A. L. Jones and S. Sung Mechanisms underlying epigenetic regulation in Arabidopsis thaliana. Integrative and comparative biology, 54 (2014), 61-7. |
| [40] | E. J. Kim, X. Ma and H. Cerutti Gene silencing in microalgae: Mechanisms and biological roles. Bioresource technology, (2014), in press. |
| [41] | P. T. West, Q. Li, L. Ji, S. R. Eichten, J. Song, M. W. Vaughn, R. J. Schmitz and N. M. Springer Genomic distribution of H3K9me2 and DNA methylation in a maize genome. PloS One, 9 (2014), e105267. |
| [42] | M. Y. Kim and D. Zilberman DNA methylation as a system of plant genomic immunity. Trends in plant science, 19 (2014), 320-6. |
| [43] | D. T. Ge and P. D. Zamore Small RNA-directed silencing: the y finds its inner fission yeast? Current biology, 23 (2013), R318-20. |
| [44] | Z. Guo, W. Jiang, N. Lages, W. Borcherds and D. Wang Relationship between gene duplicability and diversifiability in the topology of biochemical networks. BMC genomics, 15 (2014), 577. |
| [45] | A. N. Nguyen Ba, B. Strome, J. J. Hua, J. Desmond, I. Gagnon-Arsenault, E. L. Weiss, C. R. Landry and A. M. Moses Detecting Functional Divergence after Gene Duplication through Evolutionary Changes in Posttranslational Regulatory Sequences. PLoS computational biology, 10 (2014), e1003977. |
| [46] | A. S. Rodin, G. Gogoshin, A. Litvinenko, E. Boerwinkle Exploring Genetic Epidemiology Data with Bayesian Networks Handbook of Statistics, 28 (2012), 479-510 |
| [47] | Y. Zhang, J. Zhang and J. Shang Quantitative identification of differentially methylated loci based on relative entropy for matched case-control data. Epigenomics, 5 (2013), 631-43. |
| [48] | C. Faulk, A. Barks, K. Liu, J. M. Goodrich and D. C. Dolinoy Early-life lead exposure results in dose- and sex-specific effects on weight and epigenetic gene regulation in weanling miceu. Epigenomics, 5 (2013), 487-500. |
| [49] | J. A. Capra and D. Kostka Modeling DNA methylation dynamics with approaches from phylogenetics. Bioinformatics (Oxford, England), 30 (2014), i408-14. |
| [50] | A. Y. Chang and B. Y. Liao DNA methylation rebalances gene dosage after mammalian gene duplications. Molecular biology and evolution, 29 (2012), 133-44. |
| [51] | S. Toyoda, M. Kawaguchi, T. Kobayashi, E. Tarusawa, T. Toyama, M. Okano, M. Oda, H. Nakauchi, Y. Yoshimura, M. Sanbo, M. Hirabayashi, T. Hirayama, T. Hirabayashi and T. Yagi Developmental epigenetic modification regulates stochastic expression of clustered protocadherin genes, generating single neuron diversity. Neuron, 82 (2014), 94-108. |
| [52] | C. Lorthongpanich, L. F. Cheow, S. Balu, S. R. Quake, B. B. Knowles, W. F. Burkholder, D. Solter and D. M. Messerschmidt Single-cell DNA-methylation analysis reveals epigenetic chimerism in preimplantation embryos. Science (New York, N.Y.), 341 (2013), 1110-2. |
| [53] | H. Guo, P. Zhu, X. Wu, X. Li, L. Wen and F. Tang Single-cell methylome landscapes of mouse embryonic stem cells and early embryos analyzed using reduced representation bisulfite sequencing. Genome research, 23 (2013), 2126-35. |
| [54] | Z. S. Singer, J. Yong, J. Tischler, J. A. Hackett, A. Altinok, M. A. Surani, L. Cai and M. B. Elowitz Dynamic heterogeneity and DNA methylation in embryonic stem cells. Molecular cell, 55 (2014), 319-31. |
| [55] | J. A. Doudna and E. Charpentier Genome editing. The new frontier of genome engineering with CRISPR-Cas9. Science (New York, N.Y.), 346 (2014), 1258096. |
| 1. | Heng YANG, Jiang ZHOU , Necessary and sufficient conditions for boundedness of commutators of parametric Marcinkiewicz integrals with weighted Lipschitz functions, 2024, 25, 14549069, 277, 10.59277/PRA-SER.A.25.4.03 |
Figures(5)
Sergio Branciamore, Andrei S. Rodin, Grigoriy Gogoshin, Arthur D. Riggs. Epigenetics and Evolution: Transposons and the Stochastic Epigenetic Modification Model[J]. AIMS Genetics, 2015, 2(2): 148-162. doi: 10.3934/genet.2015.2.148
DownLoad: