In the paper, by Holder’s integral inequality, the authors establish some Hermite-Hadamard type integral inequalities for harmonically quasi-convex functions.
The following definitions for various convex functions are well known in the literature.
Definition 1.1 A function is said to be convex if
![]() |
holds for all and
.
Definition 1.2 ( 1, 2, 3) A function is said to be quasi-convex if
![]() |
holds for all and
.
Definition 1.3 ( 4) For with
and
if
![]() |
is valid for all and
, then we say that
is an
-convex function on
.
Definition 1.4 ( 5) Let with
and
. If
![]() |
is valid for all and
, then we say that
is an
-convex function on
.
Definition 1.5 ( 9) A function
is said to be a harmonically quasi-convex function on
if
![]() |
holds for all and
.
In 7, 8, the following inequalities of Hermite-Hadamard type were established.
Theorem 1.1 ( 7, Theorems 2.2 and 2.3]). Let be a differentiable mapping and
with
. Then
(i) if is convex on
, then
![]() |
(ii) if is convex on
for
, then
![]() |
Theorem 1.2 ( 8, Theorem 2.3]) Let be differentiable on
and
with
. If
is s-convex on
for
, then
![]() |
In this paper, we will create some integral inequalities of Hermite-Hadamard type for harmonically quasi-convex functions.
For creating some integral inequalities of Hermite-Hadamard type for harmonically quasi-convex functions, we need the following lemma.
Lemma 2.1 Let be differentiable on
with
, and
. If
, then
![]() |
where
Proof Integrating by part and changing variables for
yield
![]() |
Lemma 2.1 is thus proved.
Now we set off to create some integral inequalities of Hermite-Hadamard type for harmonically quasi-convex functions.
Theorem 3.1 Let be a differentiable function,
with
and
. If
is harmonically quasi-convex on
, then
![]() |
![]() |
Proof Using Lemma 2.1 and the harmonic quasi-convexity of , we have
![]() |
The proof of Theorem 3.1 is complete.
Theorem 3.2 Let be a differentiable function,
with
, and
. If
is harmonically quasi-convex on
and
, then
![]() |
Proof Since is harmonically quasi-convex on
, by Lemma 2.1 and Hölder's inequality, we have
![]() |
Theorem 3.2 is thus proved.
Theorem 3.3. Let be a differentiable function,
with
, and
. If
is harmonically quasi-convex on
for
, then
![]() |
Proof From the GA-inequality, we have
![]() |
for all . By Lemma 2.1 and the harmonic quasi-convexity of
and Hölder's inequality, we have
![]() |
![]() |
The proof of Theorem 3.3 is complete.
[1] | W. Fenchel, Convex cones, sets, and functions, Mimeographed Lectures Notes, Princeton University, Princeton, New Jersey, 1951. | ||
In article | PubMed | ||
[2] | K. L. Arrow and C. Enthovena, Quasi-concave programming, Econometrica, 1961, 29: 779-800. | ||
In article | View Article | ||
[3] | S. S. Dragomir, J. Pečaric and L. E. Persson, Some inequalities of Hadamard type, Soochow J. Math. 21 (1995), no. 3, 335-341. | ||
In article | View Article | ||
[4] | G. Toader. Some generalizations of the convexity. Proceedings of the Colloquium on Approximation and Optimization, Univ.Cluj-Napoca, Cluj-Napoca, 1985. | ||
In article | PubMed | ||
[5] | V. G. Miheşan, A generalization of the convexity, Seminar on Functional Equations, Approx. and Convex., Cluj-Napoca (Romania), 1993. | ||
In article | |||
[6] | Bo-Yan Xi, Tian-Yu Zhang, and Feng Qi. Some inequalities of Hermite-Hadamard type for m-harmonic-arithmetically convex functions. ScienceAsia, 2015, 41 (5): 357-361. | ||
In article | View Article | ||
[7] | S. S. Dragomir and R. P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 1998, 11: 91-95. | ||
In article | View Article | ||
[8] | U. S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comp., 147 (2004), 137-146. | ||
In article | View Article | ||
[9] | Tian-Yu Zhang, Ai-Ping Ji, and Feng Qi. Integral inequalities of Hermite-Hadamard type for harmonically quasi-convex functions. Proceedings of the Jangjeon Mathematical Society, 2013, 16 (3), 399-407. | ||
In article | View Article | ||
[10] | S. S. Dragomir, On some new inequalities of Hermite-Hadamard type for m-convex functions, Tamkang J. Math. 33 (2002) 45-55. | ||
In article | View Article | ||
[11] | S. S. Dragomir, G. Toader, Some inequalities for m-convex functions, Studia Univ. Babȩs-Bolyai Math. 38 (1993) 21-28. | ||
In article | |||
[12] | Bo-Yan Xi and Feng Qi. Some new integral inequalities of Hermite-Hadamard type for (log, (α,m))-convex functions on co-ordinates. Studia Universitatis Babȩs-Bolyai Mathematica, 2015, 60 (4): 509-525. | ||
In article | View Article | ||
[13] | Bo-Yan Xi and Feng Qi. Integral inequalities of Hermite-Hadamard type for ((α,m), log)-convex functions on co-ordinates. Problemy Analiza-Issues of Analysis, 2015, 22 (2): 73-92. | ||
In article | View Article | ||
[14] | Bo-Yan Xi and Feng Qi. Hermite-Hadamard type inequalities for geometrically r-convex functions. Studia Scientiarum Mathematicarum Hungarica, 2014, 51(4): 530-546. | ||
In article | View Article | ||
[15] | Bo-Yan Xi and Feng Qi. Some Hermite-Hadamard type inequalities for differentiable convex functions and applications. Hacettepe Journal of Mathematics and Statistics, 2013, 42(3): 243-257. | ||
In article | View Article | ||
[16] | Bo-Yan Xi and Feng Qi. Integral inequalities of Simpson type for logarithmically convex functions. Advanced Studies in Contemporary Mathematics, 2013, 23(4): 559-566. | ||
In article | View Article | ||
[17] | Bo-Yan Xi and Feng Qi. Hermite-Hadamard type inequalities for functions whose derivatives are of convexities. Nonlinear Functional Analysis and Applications, 2013, 18(2),: 163-176. | ||
In article | View Article | ||
Published with license by Science and Education Publishing, Copyright © 2017 Chun-Long Li, Gui-Hua Gu and Bai-Ni Guo
This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit
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[1] | W. Fenchel, Convex cones, sets, and functions, Mimeographed Lectures Notes, Princeton University, Princeton, New Jersey, 1951. | ||
In article | PubMed | ||
[2] | K. L. Arrow and C. Enthovena, Quasi-concave programming, Econometrica, 1961, 29: 779-800. | ||
In article | View Article | ||
[3] | S. S. Dragomir, J. Pečaric and L. E. Persson, Some inequalities of Hadamard type, Soochow J. Math. 21 (1995), no. 3, 335-341. | ||
In article | View Article | ||
[4] | G. Toader. Some generalizations of the convexity. Proceedings of the Colloquium on Approximation and Optimization, Univ.Cluj-Napoca, Cluj-Napoca, 1985. | ||
In article | PubMed | ||
[5] | V. G. Miheşan, A generalization of the convexity, Seminar on Functional Equations, Approx. and Convex., Cluj-Napoca (Romania), 1993. | ||
In article | |||
[6] | Bo-Yan Xi, Tian-Yu Zhang, and Feng Qi. Some inequalities of Hermite-Hadamard type for m-harmonic-arithmetically convex functions. ScienceAsia, 2015, 41 (5): 357-361. | ||
In article | View Article | ||
[7] | S. S. Dragomir and R. P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 1998, 11: 91-95. | ||
In article | View Article | ||
[8] | U. S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comp., 147 (2004), 137-146. | ||
In article | View Article | ||
[9] | Tian-Yu Zhang, Ai-Ping Ji, and Feng Qi. Integral inequalities of Hermite-Hadamard type for harmonically quasi-convex functions. Proceedings of the Jangjeon Mathematical Society, 2013, 16 (3), 399-407. | ||
In article | View Article | ||
[10] | S. S. Dragomir, On some new inequalities of Hermite-Hadamard type for m-convex functions, Tamkang J. Math. 33 (2002) 45-55. | ||
In article | View Article | ||
[11] | S. S. Dragomir, G. Toader, Some inequalities for m-convex functions, Studia Univ. Babȩs-Bolyai Math. 38 (1993) 21-28. | ||
In article | |||
[12] | Bo-Yan Xi and Feng Qi. Some new integral inequalities of Hermite-Hadamard type for (log, (α,m))-convex functions on co-ordinates. Studia Universitatis Babȩs-Bolyai Mathematica, 2015, 60 (4): 509-525. | ||
In article | View Article | ||
[13] | Bo-Yan Xi and Feng Qi. Integral inequalities of Hermite-Hadamard type for ((α,m), log)-convex functions on co-ordinates. Problemy Analiza-Issues of Analysis, 2015, 22 (2): 73-92. | ||
In article | View Article | ||
[14] | Bo-Yan Xi and Feng Qi. Hermite-Hadamard type inequalities for geometrically r-convex functions. Studia Scientiarum Mathematicarum Hungarica, 2014, 51(4): 530-546. | ||
In article | View Article | ||
[15] | Bo-Yan Xi and Feng Qi. Some Hermite-Hadamard type inequalities for differentiable convex functions and applications. Hacettepe Journal of Mathematics and Statistics, 2013, 42(3): 243-257. | ||
In article | View Article | ||
[16] | Bo-Yan Xi and Feng Qi. Integral inequalities of Simpson type for logarithmically convex functions. Advanced Studies in Contemporary Mathematics, 2013, 23(4): 559-566. | ||
In article | View Article | ||
[17] | Bo-Yan Xi and Feng Qi. Hermite-Hadamard type inequalities for functions whose derivatives are of convexities. Nonlinear Functional Analysis and Applications, 2013, 18(2),: 163-176. | ||
In article | View Article | ||