Abstract
By Hölder’s integral inequality, the authors establish some Hermite–Hadamard type integral inequalities for n-times differentiable and geometrically quasi-convex functions.
Similar content being viewed by others
Background
Let I be an interval on \({\mathbb {R}}=(-\infty ,\infty )\). A function \(f:I\rightarrow {\mathbb {R}}\) is said to be convex if
for \(x,y\in I\) and \(\lambda \in [0,1]\). If the inequality (1) reverses, then f is said to be concave on I.
A function \(f:I\subseteq {\mathbb {R}}_+=(0,\infty )\rightarrow {\mathbb {R}}_+\) is said to be geometrically convex on I if
for \(x,y\in I\) and \(\lambda \in [0,1]\).
One of the most famous inequalities for convex functions is Hermite–Hadamard’s inequality: if \(f:I\subseteq {\mathbb {R}}\rightarrow {\mathbb {R}}\) is convex on an interval I of real numbers and \(a,b\in I\) with \(a<b\), then
if f is concave on I, then the inequality (2) is reversed.
We now collect several Hermite–Hadamard type integral inequalities as follows.
Theorem 1
(Dragomir and Agarwal 1998) Let \(f:I^\circ \subseteq {\mathbb {R}}\rightarrow {\mathbb {R}}\) be a differentiable mapping on \(I^\circ \) and \(a,b\in I^\circ \) with \(a<b\). If \(|f'|\) is convex on [a, b], then
Theorem 2
(Xi and Qi 2013) Let \(f:I \subseteq {\mathbb {R}}_+\rightarrow {\mathbb {R}}\) be a differentiable function on \(I^\circ \) and \(a,b\in I^\circ \) with \(a<b\). If \(|f'|\) is geometrically convex on [a, b], then
where
for \(u,v>0\) and \(u\ne v\) is called the logarithmic mean.
Theorem 3
(Dragomir and Agarwal 1998) Let \(f:I^\circ \subseteq {\mathbb {R}}\rightarrow {\mathbb {R}}\) be a differentiable mapping on \(I^\circ \) and \(a,b\in I^\circ \) with \(a<b\). If \(|f'|^q\) for \(q\ge 1\) is convex on [a, b], then
and
Theorem 4
(Kirmaci 2004) Let \(f:I\subseteq {\mathbb {R}}\rightarrow {\mathbb {R}}\) be differentiable on \(I^\circ \) and \(a,b\in I\) with \(a<b\). If \(|f'|^{p/(p-1)}\) for \(p>1\) is convex on [a, b], then
Corresponding to the concept of geometrically convex functions, the geometrically quasi-convex functions were introduced in Qi and Xi (2014) as follows.
Definition 1
(Definition 2.1 Qi and Xi 2014) A function \(f:I\subseteq {\mathbb {R}}_+\rightarrow {\mathbb {R}}_0=[0,\infty )\) is said to be geometrically quasi-convex function on I if
for \(x,y\in I\) and \(\lambda \in [0,1]\).
In Qi and Xi (2014), some integral inequalities of Hermite–Hadamard type for geometrically quasi-convex functions were established.
In recent years, some other kinds of Hermite–Hadamard type inequalities were generated. For more systematic information, please refer to Bai et al. (2012), Pearce and Pečarić (2000), Pečarić and Tong (1991), Wang and Qi (2013), Wang et al. (2012), Xi et al. (2012) and related references therein.
The aim of this paper is to find more integral inequalities of Hermite–Hadamard type for n-times differentiable and geometrically quasi-convex functions.
A Lemma
In order to obtain our main results, we need the following Lemma.
Lemma 1
(Wang and Shi 2016) For \(n\in {\mathbb {N}}\), let \(f:I\subseteq {\mathbb {R}}_+\rightarrow {\mathbb {R}}\) be a n-times differentiable function on \(I^\circ \) and \(a,b\in I\) with \(a< b\). If \(f^{(n)}\in L_1([a,b])\), then
Remark 1
Under the conditions of Lemma 1, taking \(n=1\), we obtain
which can be found in Zhang et al. (2013).
Inequalities for geometrically quasi-convex functions
Now we start out to establish some new Hermite–Hadamard type inequalities for n-times differentiable and geometrically quasi-convex functions.
Theorem 5
For \(n\in {\mathbb {N}}\) , suppose that \(f:I\subseteq {\mathbb {R}}_+\rightarrow {\mathbb {R}}\) is a n-times differentiable function on \(I^\circ \) , that \(f^{(n)}\in L_1([a,b])\) , and that \(a,b\in I\) with \(a<b\) . If \(\bigl |f^{(n)}\bigr |^q\) is geometrically quasi-convex on [a, b] for \(q\ge 1\) , then
Proof
By the geometric quasi-convexity of \(\bigl |f^{(n)}\bigr |^q\), Lemma 1, and Hölder’s inequality, one has
Theorem 5 is thus proved.\(\square \)
Corollary 1
Under the assumptions of Theorem 5, if \(q=1\), then
Theorem 6
For \(n\in {\mathbb {N}}\), suppose that \(f:I\subseteq {\mathbb {R}}_+\rightarrow {\mathbb {R}}\) is a n-times differentiable function on \(I^\circ \), that \(f^{(n)}\in L_1([a,b])\), and that \(a,b\in I\) with \(a<b\). If \(\bigl |f^{(n)}\bigr |^q\) is geometrically quasi-convex on [a, b] for \(q>1\), then
for \(0\le m, r\le (n+1)q\).
Proof
From the geometric quasi-convexity of \(\bigl |f^{(n)}\bigr |^q\), Lemma 1, and Hölder’s inequality, we have
The proof of Theorem 6 is complete. \(\square \)
Corollary 2
Under the conditions in Theorem 6,
-
1.
if \(m=r=0\), then
$$\begin{aligned}&\Biggl |\sum _{k=1}^{n}\frac{(-1)^{k-1}}{k!}\bigl [b^kf^{(k-1)}(b)-a^{k}f^{(k-1)}(a)\bigr ] -\int _{a}^{b}f(x){{\mathrm{d}}}x \Biggl |\\&\quad \le \frac{\ln b-\ln a}{n!}\Bigl [L\Bigl (a^\frac{q(n+1)}{q-1}, b^\frac{q(n+1)}{q-1}\Bigr )\Bigr ]^{1-1/q}\sup \bigl \{\bigr |f^{(n)}(a)\bigr |,\bigr |f^{(n)}(b)\bigr |\bigr \}; \end{aligned}$$ -
2.
if \(m=r=q(n+1)\), then
$$\begin{aligned}&\Biggl |\sum _{k=1}^{n}\frac{(-1)^{k-1}}{k!}\bigl [b^kf^{(k-1)}(b)-a^{k}f^{(k-1)}(a)\bigr ] -\int _{a}^{b}f(x){{\mathrm{d}}}x \Biggl |\\&\quad \le \frac{\ln b-\ln a}{ n!}\bigl [L\bigl (a^{q(n+1)}, b^{q(n+1)}\bigr )\bigr ]^{1/q}\sup \bigl \{\bigr |f^{(n)}(a)\bigr |,\bigr |f^{(n)}(b)\bigr |\bigr \}; \end{aligned}$$ -
3.
if \(m=0\) and \(r=q(n+1)\), then
$$\begin{aligned}&\Biggl |\sum _{k=1}^{n}\frac{(-1)^{k-1}}{k!}\bigl [b^kf^{(k-1)}(b)-a^{k}f^{(k-1)}(a)\bigr ] -\int _{a}^{b}f(x){{\mathrm{d}}}x \Biggl |\le \frac{\ln b-\ln a}{n!}\\&\quad \times \Bigl [L\Bigl (a^\frac{q(n+1)}{q-1}, 1\Bigr )\Bigr ]^{1-1/q}\bigl [L\bigl (1, b^{q(n+1)}\bigr )\bigr ]^{1/q}\sup \bigl \{\bigr |f^{(n)}(a)\bigr |,\bigr |f^{(n)}(b)\bigr |\bigr \}; \end{aligned}$$ -
4.
if \(m=n+1\) and \(r=q(n+1)\), then
$$\begin{aligned}&\Biggl |\sum _{k=1}^{n}\frac{(-1)^{k-1}}{k!}\bigl [b^kf^{(k-1)}(b)-a^{k}f^{(k-1)}(a)\bigr ] -\int _{a}^{b}f(x){{\mathrm{d}}}x \Biggl |\le \frac{\ln b-\ln a}{n!}\\&\quad \times \bigl [L\bigl (a^{n+1}, 1\bigr )\bigr ]^{1-1/q}\bigl [L\bigl (a^{n+1}, b^{q(n+1}\bigr )\bigr ]^{1/q}\sup \bigl \{\bigr |f^{(n)}(a)\bigr |,\bigr |f^{(n)}(b)\bigr |\bigr \}; \end{aligned}$$ -
5.
if \(m=q(n+1)\) and \(r=0\), then
$$\begin{aligned}&\Biggl |\sum _{k=1}^{n}\frac{(-1)^{k-1}}{k!}\bigl [b^kf^{(k-1)}(b)-a^{k}f^{(k-1)}(a)\bigr ] -\int _{a}^{b}f(x){{\mathrm{d}}}x \Biggl |\le \frac{\ln b-\ln a}{n!}\nonumber \\&\quad \times \Bigl [L\Bigl (1, b^\frac{q(n+1)}{q-1}\Bigr )\Bigr ]^{1-1/q}\bigl [L\bigl (a^{q(n+1}, 1\bigr )\bigr ]^{1/q}\sup \bigl \{\bigr |f^{(n)}(a)\bigr |,\bigr |f^{(n)}(b)\bigr |\bigr \}; \end{aligned}$$ -
6.
if \(m=q(n+1)\) and \(r=n+1\), then
$$\begin{aligned}&\Biggl |\sum _{k=1}^{n}\frac{(-1)^{k-1}}{k!}\bigl [b^kf^{(k-1)}(b)-a^{k}f^{(k-1)}(a)\bigr ] -\int _{a}^{b}f(x){{\mathrm{d}}}x \Biggl |\le \frac{\ln b-\ln a}{n!}\\&\quad \times \bigl [L\bigl (1, b^{n+1}\bigr )\bigr ]^{1-1/q}\bigl [L\bigl (a^{q(n+1)}, b^{n+1}\bigr )\bigr ]^{1/q} \sup \bigl \{\bigr |f^{(n)}(a)\bigr |,\bigr |f^{(n)}(b)\bigr |\bigr \}. \end{aligned}$$
Conclusion
Our main results in this paper are those integral inequalities of Hermite–Hadamard type in Theorems 5 and 6 and Corollaries 1 and 2.
References
Bai S-P, Wang S-H, Qi F (2012) Some Hermite-Hadamard type inequalities for n-time differentiable (α, m)-convex functions. J Inequal Appl 267:11. doi:10.1186/1029-242X-2012-267
Dragomir SS, Agarwal RP (1998) Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula. Appl Math Lett 11(5):91–95. doi:10.1016/S0893-9659(98)00086-X
Kirmaci US (2004) Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula. Appl Math Comp 147(1):137–146. doi:10.1016/S0096-3003(02)00657-4
Pearce CEM, Pečarić JE (2000) Inequalities for differentiable mappings with application to special means and quadrature formulae. Appl Math Lett 13(2):51–55. doi:10.1016/S0893-9659(99)00164-0
Pečarić JE, Tong YL (1991) Convex functions, partial ordering and statistical applications. Academic Press, New York
Qi F, Xi B-Y (2014) Some Hermite-Hadamard type inequalities for geometrically quasi-convex functions. Proc Indian Acad Sci Math Sci 124(3):333–342. doi:10.1007/s12044-014-0182-7
Wang S-H, Qi F (2013) Inequalities of Hermite-Hadamard type for convex functions which are n-times differentiable. Math Inequal Appl 16(4):1269–1278. doi:10.7153/mia-16-97
Wang S-H, Shi X-T (2016) Hermite-Hadamard type inequalities for n-times differentiable and GA-convex functions with applications to means. J Anal Number Theory 4(1):15–22. doi:10.18576/jant/040103
Wang S-H, Xi B-Y, Qi F (2012) On Hermite-Hadamard type inequalities for (α, m)-convex functions. Int J Open Probl Comput Sci Math 5(4):47–56. doi:10.12816/0006138
Xi B-Y, Bai R-F, Qi F (2012) Hermite-Hadamard type inequalities for the m- and (α, m)-geometrically convex functions. Aequ Math 84(3):261–269. doi:10.1007/s00010-011-0114-x
Xi B-Y, Qi F (2013) Hermite–Hadamard type inequalities for functions whose derivatives are of convexities. Nonlinear Funct Anal Appl 18(2):163–176
Zhang T-Y, Ji A-P, Qi F (2013) Some inequalities of Hermite–Hadamard type for GA-convex functions with applications to means. Matematiche 68(1):229–239. doi:10.4418/2013.68.1.17
Authors’ contributions
JZ, FQ, G-CX and Z-LP contributed equally to the manuscript. All authors read and approved the final manuscript.
Acknowledgements
This work was partially supported by the National Natural Science Foundation of China under Grant Nos. 61163034 and 61373067 and by the Inner Mongolia Autonomous Region Natural Science Foundation Project under Grant No. 2015MS0123, China. The authors appreciate anonymous referees for their valuable comments on and careful corrections to the original version of this paper.
Competing interests
The authors declare that they have no competing interests.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Zhang, J., Qi, F., Xu, GC. et al. Hermite–Hadamard type inequalities for n-times differentiable and geometrically quasi-convex functions. SpringerPlus 5, 524 (2016). https://doi.org/10.1186/s40064-016-2083-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s40064-016-2083-y