Integral Majorization Type Inequalities for the Functions in the Sense of Strong Convexity

Zaheer Ullah, Syed; Khan, Muhammad Adil; Khan, Zareen Abdulhameed; Chu, Yu-Ming

doi:https://doi.org/10.1155/2019/9487823

Journal of Function Spaces

On this page

Abstract Introduction Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Special Issue

Operator Methods in Approximation Theory

View this Special Issue

Research Article | Open Access

Volume 2019 | Article ID 9487823 | https://doi.org/10.1155/2019/9487823

Integral Majorization Type Inequalities for the Functions in the Sense of Strong Convexity

Syed Zaheer Ullah,¹Muhammad Adil Khan,¹Zareen Abdulhameed Khan,²and Yu-Ming Chu³

Guest Editor: Vita Leonessa

Received24 Jan 2019

Accepted27 Mar 2019

Published02 May 2019

Abstract

In this article, we establish several integral majorization type and generalized Favard’s inequalities for the class of strongly convex functions. Our results generalize and improve the previous known results.

1. Introduction

It is well known that convex functions are a class of important functions in the fields of mathematics and other natural sciences; they have been studied for more than one hundred years. In recent years there is a growing interest in generalized convex functions (such as quasi-convex function [1], strongly convex function [2–4], -convex function [5], approximately convex function [6], logarithmically convex function [7, 8], midconvex function [9], pseudo-convex function [10], -convex function [11], -convex function [12], -convex function [13], delta-convex function [14], Schur convex function [15–21], and other convex functions [22–29]) among the researchers of applied mathematics due to the fact that mathematical models with these functions are more suitable to describe problems of the real world than models using conventional convex functions. Recently, a large number of remarkable results and applications for the generalized convex functions can be found in the literature [30–49].

In the article, our focus is on the integral majorization type inequalities for the strongly convex functions.

Definition 1. Let be a real-valued function defined on the interval and a positive real number. Then is said to be strongly convex with modulus if the inequalityholds for all and . From (1) we clearly see that

The following Lemma 2 for strongly convex function is given in [2] (see also [50, Proposition 1.1.2]).

Lemma 2. A real-valued function is a strongly convex function with modulus if and only if the function defined by is a convex function.

Every strongly convex function is convex, but the converse is not true in general. Strongly convex functions have been utilized for showing the convergence of a gradient type algorithm for minimizing a function. They play a significant role in mathematical economics, approximation theory, and optimization theory; many applications and properties for strongly functions can be found in [2–4, 13, 30].

Next we are going to present some basic theories of majorization.

There is a natural description of the indefinite notion that the entries of -tuple are more nearly equal, or less spread out than, to the entries of -tuple . The applicable assertion is that majorizes ; it means that the sum of largest entries of does not exceed the sum of largest entries of for all with equality for . That is, let and be two real tuples and letbe their ordered entries. Then the tuple is said to majorize (or is said to be majorized by ), in symbol , if holds for and

The theory of majorization is a very significant topic in mathematics; a remarkable and complete reference on the majorization subject is the book by Olkin and Marshall [51]. For example, the theory of majorization is an essential tool that permits us to transform nonconvex complicated constrained optimization problems that involve matrix valued variables into simple problems with scalar variables that can be easily solved [52–55].

The definition of majorization for integrable functions can be stated as follows (see [7]).

Definition 3. Let and be two decreasing real-valued integrable functions on the interval . Then is said to majorize (or is said to be majorized by ), in symbol, , if the inequalityholds for all and

Theorem 4 (See [56]). Let and be two continuous and increasing real-valued functions defined on , and let be a bounded variation function. Then the following statements are true. (a)If for all and then holds for every continuous convex function .(b)If (8) and (9) hold, then (10) holds for every continuous increasing convex function .

Theorem 5 (See [57]). Let be a convex function, , and be three positive and integrable functions defined on such that for all and Then the following statements are true: (a)If is decreasing on , then(b)If is increasing on , then

Let , be a positive and continuous concave function defined on , and let be a convex function defined on with Then Favard [58] proved that the inequalities and hold.

The main purpose of the article is to establish several integral majorization type and generalized Favard’s inequalities for strongly convex functions.

2. Main Results

Theorem 6. Let , be a continuous strongly convex function with modulus , and let , , and be three positive and integrable functions defined on such thatfor all andThen the following statements are true. (a)If is decreasing on , then we have (b)If is increasing on , then one has

Proof. (a) Let and . Then it follows from (2) and the proof of Lemma 2 given in [57] that Let . Then (18) and (19) lead to for all , , and Since is decreasing on , therefore inequality (20) can be deduced easily from the above inequality. Similarly, we can prove part (b) for increasing function defined on .

Theorem 7. Suppose that all the assumptions of Theorem 6 hold. Then the following statements are true. (a)If is decreasing on , then(b)If is increasing on , then

Proof. Since is a strongly convex function with modulus , therefore is a convex function, and inequalities (24) and (25) follow easily from the convexity of the function and Lemma 2 given in [57].

Remark 8. Inequalities (13) and (14) can be obtained by (24) and (25) immediately.

Remark 9. Generally, the assumptions of both the functions and are monotonic in majorization theorem, but in Theorems 6 and 7 we only need one of the functions and to be monotonic.

Theorem 10. Suppose that is a continuous strongly convex function with modulus , and , , and are three integrable functions on . If and are nondecreasing (nonincreasing) functions on and , then

Proof. Since is a strongly convex function, therefore using (2) for and , we haveIt follows from the Čebyšev inequality [59] thatTherefore, inequality (26) follows from (27) and (28).

Making use of the similar idea as in the proof of Theorem 10, we can obtain the following Theorem 11 immediately.

Theorem 11. Suppose that is a continuous strongly convex function with modulus , and , , and are three integrable functions on . If and are nondecreasing (nonincreasing) functions on , and , then inequality (26) holds.

Theorem 12. The inequalityholds if all the assumptions of Theorem 10 are satisfied.

Proof. Since is a strongly convex function with modulus , therefore is a convex function and inequality (29) can be deduced by applying this convex function in Theorem 6 of [59].

Using strongly convex function we can give an extension of [60, Theorem 2] in the following form.

Theorem 13. Let be two functions such that is a strictly increasing and is strongly convex with modulus , , , and being three positive and integrable functions on such that for all and Then the following statements are true. (a)If is decreasing on , then inequality (20) holds.(b)If is increasing on , then inequality (21) holds.

Proof. We clearly see that it is sufficient to prove the case of , but this case is already proved in Theorem 6.

Similarly, we have Theorem 14 as follows.

Theorem 14. Suppose that all the assumptions of Theorem 13 are satisfied. Then the following statements are true. (a)If is decreasing on , then inequality (24) holds.(b)If is increasing on , then inequality (25) holds.

The following Lemma 15 was given in [57].

Lemma 15. Let be a positive integrable function and let be an increasing function on ; thenIf is a decreasing function on , then inequality (32) holds in reverse directions.

Lemma 16. Let be a real-valued function defined on . Then the following statements are true. (i)If is a strongly concave function with modulus , then(a)the function is decreasing on ;(b)the function is increasing on .(ii)If is a strongly convex function with modulus , then(c)the function is increasing on if ;(d)the function is decreasing on if .

Proof. (i) Suppose that is a strongly concave function with modulus .
(a) To show that the function is decreasing on , in fact, for we have which shows that the function is decreasing on .
(b) To show that the function is increasing on , in fact, for we have which shows that the function is increasing on .
(ii) Suppose that is a strongly convex function with modulus .
(c) Since , by similar method of (a) we can easily prove that the function is increasing on .
(d) Since , by similar method of (b) we can prove that the function is decreasing on .

Next, we establish several Favard type inequalities for strongly convex functions.

Theorem 17. Let be a strongly concave function with modulus on such that is a positive increasing function, let be a strongly convex function with modulus on , , andThenIf is a strongly convex function with modulus on such that is a positive increasing function and , then the reverse inequality in (36) holds.
Let be a strongly concave function with modulus on such that is a positive decreasing function, let be a strongly convex function with modulus on , , andThenIf is a strongly convex function with modulus on such that is a positive decreasing function and , then the reverse inequality in (38) holds.

Proof. (a) From Lemma 16(a) we know that the function is decreasing; then using Lemma 15 to the functions and , we obtainIt follows from (35) that inequality (39) can be rewritten asfor all .
As is an increasing function, and by use of Theorem 6(b), we haveNote that Therefore, we get If is a strongly convex function with modulus on such that is a positive increasing function and , then the reverse inequality in (36) can be proved by using a similar method as in the proof of part (a) and Lemma 16(c).
(b) From Lemma 16(b) we know that the function is increasing; then using Lemma 15 to the functions and , we obtainFrom (37) we clearly see that inequality (44) can be rewritten as for all .
As is decreasing function, and by using Theorem 6(a) we have Note that Therefore, If is a strongly convex function with modulus on such that is a positive decreasing function and , then the reverse inequality in (38) can be proved by using a similar method as in the proof of part (b) and Lemma 16(d).

Theorem 18. The following statements are true under the assumptions of Theorem 17. (a)If is a positive increasing function, then(b)If is a positive decreasing function, then

Proof. (a) We clearly see that is convex function due to being a strongly convex function with modulus . Therefore, inequality (49) follows easily from [57, Theorem 1(i)] and the strong convexity of together with the fact that is a strongly concave function with modulus on such that is positive increasing function.
(b) Similarly, inequality (50) follows easily from [57, Theorem 1(ii)] and the strong convexity of together with the fact that is a strongly concave function with modulus on such that is positive decreasing function.

Theorem 19. Let be an increasing function on , let be a decreasing function on , let , , and be three positive functions on , and let and be integrable on such thatAnd let be a strongly convex function with modulus . Then the inequality holdsfor all .

Proof. From and (51), applying Lemma 15 to the function and the decreasing function we get Since is increasing, therefore by using Theorem 6 we have

Theorem 20. Let be a strongly convex function with modulus . Then the inequalityholds for all if all the assumptions of Theorem 19 are satisfied.

Proof. We clearly see that is a convex function due to being a strongly convex function with modulus . Therefore, inequality (55) follows easily from [60, Theorem 3] and the convexity of the function .

Remark 21. Clearly, [60, Theorem 3] can be deduced from (52) due to or from (55) due to for convex function .

The following Theorem 22 is an extension of Theorem 19.

Theorem 22. Let be a continuous strongly convex function with modulus , ( is a nonnegative real number), and two positive integrable functions on , , and . Then the following statements are true. (a)If is increasing on and is decreasing on , then(b)If is increasing on and is increasing on , then

Proof. (a) Let ; then applying Lemma 15 to the function and the decreasing function , we have Therefore, inequality (58) follows from Theorem 6 and the fact that is an increasing function on .
(b) Let ; then applying Lemma 15 to the function and the increasing function , we get Therefore, inequality (59) follows from Theorem 6 and the fact that is an increasing function on .

Theorem 23. The following statements are true under the assumptions of Theorem 22. (a)If is an increasing function on and is a decreasing function on , then(b)If is increasing on and is an increasing function on , then

Proof. We clearly see that is a convex function due to being a strongly convex function with modulus . Therefore, inequalities (62) and (63) follow from [61, Theorem 2.3] and the convexity of the function .

Remark 24. Clearly, Theorem 2.3(1) and Theorem 2.3(2) given in [57] can be deduced by (62) and (63), respectively.

Remark 25. Theorem 22 is an extension of Favard’s inequality given in Theorem 17. Indeed, let be a strongly convex function with modulus , then is also a strongly convex function with modulus for any . Substituting in (58), one hasSince is a strongly concave function with modulus on such that is a positive increasing function, taking and using (35) and in (64), we obtain the Favard’s inequalities given in Theorem 17.

Remark 26. From (64) we can easily obtain Remark 2.4 given in [61] due to

For an application of Theorem 22, we get Corollary 27 as follows.

Corollary 27. Let , , , and , , , , and be stated as in Theorem 22. Then the following statements are true. (a)If is increasing on and is decreasing on , then(b)If is increasing on and is an increasing function on , then

Proof. (a) Since is increasing on and is a decreasing function on , using (58) given in Theorem 22 and substituting , we get Therefore, inequality (66) follows from (68).
(b) Since is increasing on and is an increasing function on , using (59) given in Theorem 22 and substituting we get inequality (67).

Remark 28. Let , , and be a strongly concave function with modulus on such that is a positive increasing function. Then inequality (66) leads to the classical Favard’s inequality for strongly convex functions with modulus :

Remark 29. From (69) we get the classical Favard’s inequality given in [58] due to

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the Natural Science Foundation of China (Grant nos. 61673169, 11301127, 11701176, 11626101, 11601485).

References

B. de Finetti, “Sulle stratificazioni convesse,” Annali di Matematica Pura ed Applicata, vol. 30, no. 4, pp. 173–183, 1949.
View at: Publisher Site | Google Scholar | MathSciNet
N. Merentes and K. Nikodem, “Remarks on strongly convex functions,” Aequationes Mathematicae, vol. 80, no. 1-2, pp. 193–199, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
Y.-Q. Song, M. Adil Khan, S. Zaheer Ullah, and Y.-M. Chu, “Integral inequalities involving strongly convex functions,” Journal of Function Spaces, vol. 2018, Article ID 6595921, 8 pages, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
W. Zhou and X. Chen, “On the convergence of a modified regularized Newton method for convex optimization with singular solutions,” Journal of Computational and Applied Mathematics, vol. 239, pp. 179–188, 2013.
View at: Google Scholar
M. Adil Khan, Y.-M. Chu, T. U. Khan, and J. Khan, “Some new inequalities of Hermite-Hadamard type for s-convex functions with applications,” Open Mathematics, vol. 15, pp. 1414–1430, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
D. H. Hyers and S. M. Ulam, “Approximately convex functions,” Proceedings of the American Mathematical Society, vol. 3, pp. 821–828, 1952.
View at: Publisher Site | Google Scholar | MathSciNet
J. E. Pečarič, F. Proschan, and Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, vol. 187 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1992.
View at: MathSciNet
T.-H. Zhao, Y.-M. Chu, and Y.-P. Jiang, “Monotonic and logarithmically convex properties of a function involving gamma functions,” Journal of Inequalities and Applications, vol. 2009, Article ID 728612, 13 pages, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
J. L. W. V. Jensen, “Om konvekse Funktioner og Uligheder mellem Middelværdier,” Nyt tidsskrift for matematik. B, vol. 16, pp. 49–68, 1905.
View at: Google Scholar
O. L. Mangasarian, “Pseudo-convex functions,” SIAM Journal on Control and Optimization, vol. 3, pp. 281–290, 1965.
View at: Google Scholar | MathSciNet
S. S. Dragomir, “Inequalities of Hermite-Hadamard type for φ-convex functions,” http://rgmia.org/papers/v16/v16a87.pdf.
View at: Google Scholar
S. S. Dragomir, “Inequalities of Hermite-Hadamard type for λ-convex functions on linear spaces,” http://rgmia.org/papers/v17/v17a13.pdf.
View at: Google Scholar
S. Varošanec, “On -convexity,” Journal of Mathematical Analysis and Applications, vol. 326, no. 1, pp. 303–311, 2007.
View at: Publisher Site | Google Scholar | MathSciNet
T. Rajba, “On strong delta-convexity and Hermite-Hadamard type inequalities for delta-convex functions of higher order,” Mathematical Inequalities & Applications, vol. 18, no. 1, pp. 267–293, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
H. Hu and L. Liu, “Weighted inequalities for a general commutator associated to a singular integral operator satisfying a variant of Hörmander’s condition,” Mathematical Notes, vol. 101, no. 5-6, pp. 830–840, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
X. Zhou, “Weighted sharp function estimate and boundedness for commutator associated with singular integral operator satisfying a variant of Hörmander’s condition,” Journal of Mathematical Inequalities, vol. 9, no. 2, pp. 587–596, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
C. Huang, Z. Yang, T. Yi, and X. Zou, “On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities,” Journal of Differential Equations, vol. 256, no. 7, pp. 2101–2114, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
C. Huang, S. Guo, and L. Liu, “Boundedness on Morrey space for Toeplitz type operator associated to singular integral operator with variable Calderón-Zygmund kernel,” Journal of Mathematical Inequalities, vol. 8, no. 3, pp. 453–463, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
D. Lu, “Some sharp inequalities for multilinear integral operators,” Journal of Inequalities and Applications, vol. 2013, article 445, 14 pages, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
X. Zhou, “Some sharp function estimates for vector-valued multilinear integral operator,” Miskolc Mathematical Notes, vol. 14, no. 1, pp. 373–387, 2013.
View at: Publisher Site | Google Scholar
Z.-G. Wang, G.-W. Zhang, and F.-H. Wen, “Properties and characteristics of the Srivastava-Khairnar-More integral operator,” Applied Mathematics and Computation, vol. 218, no. 15, pp. 7747–7758, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
H. Wang, W.-M. Qian, and Y.-M. Chu, “Optimal bounds for Gaussian arithmetic-geometric mean with applications to complete elliptic integral,” Journal of Function Spaces, vol. 2016, Article ID 3698463, 6 pages, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
H.-H. Chu, Z.-H. Yang, Y.-M. Chu, and W. Zhang, “Generalized Wilker-type inequalities with two parameters,” Journal of Inequalities and Applications, vol. 2016, article 187, 13 pages, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
X. Zhou, “Sharp maximal function inequalities and boundedness for Toeplitz type operator associated to singular integral operator with non-smooth kernel,” Journal of Inequalities and Applications, vol. 2014, article 141, 19 pages, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
Q. Xiao and Q. Li, “Generalized lower and upper approximations in quantales,” Journal of Applied Mathematics, vol. 2012, Article ID 648983, 11 pages, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
C. Huang, C. Peng, X. Chen, and F. Wen, “Dynamics analysis of a class of delayed economic model,” Abstract and Applied Analysis, vol. 2013, Article ID 962738, 12 pages, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
M. Adil Khan, A. Iqbal, M. Suleman, and Y.-M. Chu, “Hermite-Hadamard type inequalities for fractional integrals via Green's function,” Journal of Inequalities and Applications, vol. 2018, article 161, 15 pages, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Khurshid, M. Adil Khan, and Y.-M. Chu, “Conformable integral inequaliites of the Hermite-Hadamard thpe in terms of GG- and GA-convexities,” Journal of Function Spaces, vol. 2019, Article ID 6926107, 8 pages, 2019.
View at: Publisher Site | Google Scholar | MathSciNet
M. Adil Khan, S.-H. Wu, H. Ullah, and Y.-M. Chu, “Discrete majorization type inequalities for convex functions on rectangles,” Journal of Inequalities and Applications, vol. 2019, article 16, 18 pages, 2019.
View at: Publisher Site | Google Scholar | MathSciNet
M.-K. Wang, Y.-M. Chu, and W. Zhang, “Monotonicity and inequalities involving zero-balanced hypergeometric function,” Mathematical Inequalities & Applications, vol. 22, no. 2, pp. 601–617, 2019.
View at: Publisher Site | Google Scholar | MathSciNet
J. Wu and Y. Liu, “Uniqueness results and convergence of successive approximations for fractional differential equations,” Hacettepe Journal of Mathematics and Statistics, vol. 42, no. 2, pp. 149–158, 2013.
View at: Google Scholar | MathSciNet
R.-s. Yang and Y.-h. Ou, “Inverse coefficient problems for nonlinear elliptic variational inequalities,” Acta Mathematicae Applicatae Sinica, vol. 27, no. 1, pp. 85–92, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
Z. Liu, Y. Zhang, J. Santos, and R. Ralha, “On computing complex square roots of real matrices,” Applied Mathematics Letters, vol. 25, no. 10, pp. 1565–1568, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
L. Lin and Z.-Y. Liu, “An alternating projected gradient algorithm for nonnegative matrix factorization,” Applied Mathematics and Computation, vol. 217, no. 24, pp. 9997–10002, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
W. Wang, “High order stable Runge-Kutta methods for nonlinear generalized pantograph equations on the geometric mesh,” Applied Mathematical Modelling: Simulation and Computation for Engineering and Environmental Systems, vol. 39, no. 1, pp. 270–283, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
J. Wang, X. Chen, and L. Huang, “The number and stability of limit cycles for planar piecewise linear systems of node-saddle type,” Journal of Mathematical Analysis and Applications, vol. 469, no. 1, pp. 405–427, 2019.
View at: Publisher Site | Google Scholar | MathSciNet
F. Wang and Z. Liu, “Anti-periodic fractional boundary value problems for nonlinear differential equations of fractional order,” Advances in Difference Equations, vol. 2012, article 116, 12 pages, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
Z.-H. Yang, W.-M. Qian, Y.-M. Chu, and W. Zhang, “Monotonicity rule for the quotient of two functions and its application,” Journal of Inequalities and Applications, vol. 2017, article 106, 13 pages, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
S.-L. Qiu, X.-Y. Ma, and Y.-M. Chu, “Sharp Landen transformation inequalities for hypergeometric functions, with applications,” Journal of Mathematical Analysis and Applications, vol. 474, no. 2, pp. 1306–1337, 2019.
View at: Publisher Site | Google Scholar | MathSciNet
W. Wang, “A generalized Halanay inequality for stability of nonlinear neutral functional differential equations,” Journal of Inequalities and Applications, vol. 2010, Article ID 475019, 16 pages, 2010.
View at: Publisher Site | Google Scholar
Z.-G. Wang, H. M. Srivastava, and S.-M. Yuan, “Some basic properties of certain subclasses of meromorphically starlike functions,” Journal of Inequalities and Applications, vol. 2014, article 29, 13 pages, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
J. Li, F. Liu, L. Feng, and I. Turner, “A novel finite volume method for the Riesz space distributed-order advection-diffusion equation,” Applied Mathematical Modelling: Simulation and Computation for Engineering and Environmental Systems, vol. 46, pp. 536–553, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
C. Huang and L. Liu, “Sharp function inequalities and boundness for Toeplitz type operator related to general fractional singular integral operator,” Publications de l'Institut Mathematique, vol. 92, no. 106, pp. 165–176, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
Z.-H. Yang, Y.-M. Chu, and W. Zhang, “High accuracy asymptotic bounds for the complete elliptic integral of the second kind,” Applied Mathematics and Computation, vol. 348, pp. 552–564, 2019.
View at: Publisher Site | Google Scholar | MathSciNet
T.-H. Zhao, B.-C. Zhou, M.-K. Wang, and Y.-M. Chu, “On approximating the quasi-arithmetic mean,” Journal of Inequalities and Applications, vol. 2019, article 42, 12 pages, 2019.
View at: Publisher Site | Google Scholar | MathSciNet
E. Zhu and Y. Xu, “Pathwise estimation of stochastic functional Kolmogorov-type systems with infinite delay,” Journal of Inequalities and Applications, vol. 2012, article 171, 15 pages, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
H.-H. Chu, W.-M. Qian, Y.-M. Chu, and Y.-Q. Song, “Optimal bounds for a Toader-type mean in terms of one-parameter quadratic and contraharmonic means,” Journal of Nonlinear Sciences and Applications. JNSA, vol. 9, no. 5, pp. 3424–3432, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
S. Zaheer Ullah, M. Adil Khan, and Y.-M. Chu, “Majorization theorems for strongly convex functions,” Journal of Inequalities and Applications, vol. 2019, article 58, 13 pages, 2019.
View at: Publisher Site | Google Scholar | MathSciNet
S.-H. Wu and Y.-M. Chu, “Schur m-power convexity of generalized geometric Bonferroni mean involving three parameters,” Journal of Inequalities and Applications, vol. 2019, article 57, 11 pages, 2019.
View at: Publisher Site | Google Scholar | MathSciNet
J.-B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of Convex Analysis, Springer, Berlin, Germany, 2001.
View at: Publisher Site | MathSciNet
A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and its Applications, vol. 143 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1979.
View at: MathSciNet
Y.-M. Chu, H. Wang, and T.-H. Zhao, “Sharp bounds for the Neuman mean in terms of the quadratic and second Seiffert means,” Journal of Inequalities and Applications, vol. 2014, article 299, 14 pages, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
R. Bhatia, Matrix Analysis, Springer, New York, NY, USA, 1997.
R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, Mass, USA, 1990.
View at: MathSciNet
L. Huang, Y. Li, and K. Zhao, “Geometry of alternate matrices over Bezout domains,” Algebra Colloquium, vol. 20, no. 2, pp. 197–214, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
J. E. Pečarić, “On some inequalities for functions with nondecreasing increments,” Journal of Mathematical Analysis and Applications, vol. 98, no. 1, pp. 188–197, 1984.
View at: Publisher Site | Google Scholar | MathSciNet
L. Maligranda, J. E. Pečarič, and L. E. Persson, “Weighted Favard and Berwald inequalities,” Journal of Mathematical Analysis and Applications, vol. 190, no. 1, pp. 248–262, 1995.
View at: Publisher Site | Google Scholar | MathSciNet
J. Favard, “Sur les valeurs moyennes,” Bulletin des Sciences Mathématiques, vol. 57, pp. 54–64, 1933.
View at: Google Scholar
N. S. Barnett, P. Cerone, and S. S. Dragomir, “Majorisation inequalities for Stieltjes integrals,” Applied Mathematics Letters, vol. 22, no. 3, pp. 416–421, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
J. Pečarič and S. Abramovich, “On new majorization theorems,” Rocky Mountain Journal of Mathematics, vol. 27, no. 3, pp. 903–911, 1997.
View at: Publisher Site | Google Scholar | MathSciNet
N. Latif, J. Pečarič, and I. Perič, “On majorization, Favard and Berwald inequalities,” Annals of Functional Analysis, vol. 2, no. 1, pp. 31–50, 2011.
View at: Publisher Site | Google Scholar | MathSciNet

Copyright

Copyright © 2019 Syed Zaheer Ullah et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

702

Downloads

882

Citations