On some Hermite Hadamard type inequalities for (s, QC) convex functions

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1 Wu and Qi SpringerPlus 65:49 DOI.86/s RESEARCH Open Access On some Hermite Hadamard type ineualities for s, QC convex functions Ying Wu and Feng Qi,3* *Correspondence: Department of Mathematics, School of Science, Tianjin Polytechnic University, Tianjin City 36, China Full list of author information is available at the end of the article Abstract In the paper, the authors introduce a new notion s, QC-convex function on the coordinates and establish some Hermite Hadamard type integral ineualities for s, QC -convex functions on the co-ordinates. Keywords: Convex function, s, QC-Convex function on the co-ordinates, Hermite Hadamard s integral ineuality Mathematics Subject Classification: 6A5, 6D5, 6D, 6E6, 4A55 Background Let f : I R R be a convex function and a, b I with a < b. The double ineuality a + b f b a b a f xdx f a + f b is known in the literature as Hermite Hadamard s ineuality for convex functions. Definition Dragomir and Pearce 998; Pečarić et al. 99 A function f : I R R is said to be uasi-convex QC, if f λx + λy max{f x, f y} holds for all x, y I and λ, ]. Definition Dragomir and Pearce 998 The function f : I R R is Jensen- or J-uasi-convex JQC if x + y f max{f x, f y} 3 holds for all x, y I. Definition 3 Hudzik and Maligranda 994 Let s, ]. A function f : I R R is said to be s-convex in the second sense if 6 Wu and Qi. This article is distributed under the terms of the Creative Commons Attribution 4. International License creativecommons.org/licenses/by/4./, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Wu and Qi SpringerPlus 65:49 Page of 3 f λx + λy λ s f x + λ s f y 4 holds for all x, y I and λ, ]. Definition 4 Xi and Qi 5a For some s, ], a function f : I R R is said to be extended s-convex if f λx + λy λ s f x + λ s f y 5 is valid for all x, y I and λ,. Definition 5 Dragomir ; Dragomir and Pearce A function f : =a, b]c, d] R R is said to be convex on co-ordinates on if the partial functions f y :a, b] R, f y u = f y u, y and f x :c, d] R, f x v = f x x, v 6 are convex for all x a, b and y c, d. Definition 6 A function f : =a, b]c, d] R R is said to be convex on coordinates on if the ineuality f tx + tz, λy + λw tλf x, y + t λf x, w + tλf z, y + t λf z, w 7 holds for all t, λ, ] and x, y, z, w. Definition 7 Alomari and Darus 8 A function f : =a, b]c, d] R R =, is s-convex on for some fixed s, ] if f λx + λz, λy + λw λ s f x, y + λ s f z, w 8 holds for all x, y, z, w and λ, ]. Definition 8 Özdemir et al. a, Definition 7 A function f : =a, b]c, d] R R is called a Jensen- or J-uasi-convex function on the co-ordinates on if x + z f 9, y + w max{f x, y, f z, w} holds for all x, y, z, w. Definition 9 Özdemir et al. a, Definition 5 A function f : =a, b]c, d] R R is called a uasi-convex function on the co-ordinates on if

3 Wu and Qi SpringerPlus 65:49 Page 3 of 3 f λx + λz, λy + λw max{f x, y, f z, w} holds for all x, y, z, w and λ, ]. Theorem Dragomir ; Dragomir and Pearce Theorem. Let f : =a, b]c, d] R R be convex on the co-ordinates on with a < b and c < d. Then a + b f, c + d b f x, c + d dx + d ] a + b f, y dy b a a d c c b d f x, ydydx b ad c a c b b f x, cdx+ 4 b a a a f a, c + f b, c + f a, d + f b, d. 4 f x, ddx + d d c Theorem Özdemir et al. a, Lemma 8 Every J-uasi-convex mapping f : =a, b]c, d] R R is J-uasi-convex on the co-ordinates. Theorem 3 Özdemir et al. a, Lemma 6 Every uasi-convex mapping f : =a, b]c, d] R R is uasi-convex on the coordinates. For more information on this topic, please refer to Bai et al. 6, Hwang et al. 7, Özdemir et al., a, b, c, 4, Qi and Xi 3, Roberts and Varberg 973, Sarikaya et al., Wu et al. 6, Xi et al., 5, Xi and Qi, 3, 5a, b, c and related references therein. In this paper, we introduce a new concept s, QC-convex functions on the co-ordinates on the rectangle of R and establish some new integral ineualities of Hermite Hadamard type for s, QC-convex functions on the co-ordinates. c d ] f a, ydy + f b, ydy c Definitions and Lemmas We now introduce three new definitions Definition For s, ], a function f : =a, b]c, d] R R is said to be Js, JQC-convex on the co-ordinates on with a < b and c < d, if x + z f, y + w ] max{f x, y, f x, w}+max{f z, y, f z, w} s holds for all t, λ, ] and x, y, z, w. Remark By Definitions 8 and and Lemma, we see that, for s, ] and f : R R,

4 Wu and Qi SpringerPlus 65:49 Page 4 of 3. If f : R is a J-uasi-convex function on the co-ordinates on, then f is a Js, JQC-convex function on the co-ordinates on ;. Every J-uasi-convex function f : R is a Js, JQC-convex function on the coordinates on. Definition A function f : =a, b]c, d] R R is called s, JQC-convex on the co-ordinates on with a < b and c < d, if f tx + tz, y + w t s max{f x, y, f x, w}+ t s max{f z, y, f z, w} holds for all t,, x, y, z, w, and some s, ]. Definition For some s, ], a function f : =a, b]c, d] R R is called s, QC-convex on the co-ordinates on with a < b and c < d, if f tx + tz, λy + λw t s max{f x, y, f x, w}+ t s max{f z, y, f z, w} 3 is valid for all t,, λ, ], and x, y, z, w. Remark For s, ] and f : R R,. If taking λ = and t = λ = in 3, then Js, JQC s, JQC s, QC;. If f : R is a s-convex function on, then f is an s, QC-convex function on the co-ordinates on. Remark 3 Considering Definitions 9 and and Lemma, for s, ] and f : R R,. If f : R is a uasi-convex function on the co-ordinates on, then it is an s, QC-convex function on the co-ordinates on ;. Every uasi-convex function f : R is an s, QC-convex function on the coordinates on. Lemma Latif and Dragomir If f : =a, b]c, d] R R has partial derivatives and f L with a < b and c < d, then f ; a, b, c, d b ad c b a b a = b ad c f b d a c x, c + d a + b f x, ydydx + f dx d c d c, c + d a + b f, y dy K t, λ,

5 Wu and Qi SpringerPlus 65:49 Page 5 of 3 where tλ, t, λ, ] ],, t λ, t, λ, K t, λ = ], ], t λ, t, λ, ], ], t λ, t, λ, ], ]. 4 Lemma Let r and >. Then / u r du = / u r du = r+ r + 5 and K t, λ / = /. 4 6 where K t, λ is defined by 4. Proof This follows from a straightforward computation. Some integral ineualities of Hermite Hadamard type In this section, we will establish Hermite Hadamard type integral ineualities for s, QC -convex functions on the co-ordinates on rectangle from the plane R. Theorem 4 Let f : =a, b]c, d] R R have partial derivatives and f L. If f is an s, QC-convex function on the co-ordinates on with a < b and c < d for some s, ] and, then. When s, ], / b ad c 8 s+ s+s+ { s + M a, c, d + s+ s 3M b, c, d ] / + s+ s 3M a, c, d + s + M b, c, d ] } / ; 7. When s =, b ad c { M a, c, d + ln M b, c, d ] / 8 + ln M a, c, d + M b, c, d ] / } ; where { M u, c, d = max f u, c, f u, d }. 8 9

6 Wu and Qi SpringerPlus 65:49 Page 6 of 3 Proof By Lemma and Hölder s integral ineuality, we have f ; a, b, c, d b ad c K t, λ / b ad c K t, λ { ] / / / tλ ] / / + / t λ ] / + / tλ / ] + / / t λ / }. When s, ], using the co-ordinated s, QC-convexity of f and by Lemma, we obtain / / tλ { / / λdλ t s+ max, f a, c } f a, d { + t t s max, f b, c }] f b, d dt = s + M a, c, d + s+ s 3M b, c, d ]. s+5 s+s+ Similarly, we also have / / t λ s + M a, c, d + s+ s 3M b, c, d ], s+5 s+s+ / / tλ s+5 s+s+ s+ s 3M a, c, d + s + M b, c, d ], 3 / / t λ s+5 s+s+ s+ s 3M a, c, d + s + M b, c, d ]. 4 Applying ineualities to 4 into the ineuality yields

7 Wu and Qi SpringerPlus 65:49 Page 7 of 3 b ad c 6 { s+5 s + s + / s + M a, c, d + s+ s 3M b, c, d] / + s+ s+5 s 3M a, c, d + s + M b, c, d ]] / } s + s + b ad c / = 8 s+ s + s + { s + M a, c, d + s+ s 3M b, c, d ] / + s+ s 3M a, c, d + s + M b, c, d ] / }. When s =, similar to the proof of ineualities to 4, we can write / / tλ 4 M a, c, d + ln M b, c, d ], / / t λ 4 M a, c, d + ln M b, c, d ], 5 6 / / tλ 4 ln M a, c, d + M b, c, d ], 7 / / t λ 4 ln M a, c, d + M b, c, d ]. 8 Substituting ineualities 5 to 8 into leads to the ineuality 8. Theorem 4 is thus proved. Corollary Under the conditions of Theorem 4,. If = and s, ], then b ad cs+ s+3 s + s + { f a, c max, } { f a, d f b, c + max, }] f b, d ;. If = and s =, then b ad c ln 4 { f a, c max, } { f a, d f b, c + max, }] f b, d.

8 Wu and Qi SpringerPlus 65:49 Page 8 of 3 Corollary Under the conditions of Theorem 4,. If s =, then b ad c / 4 4 { f a, c max f a, d, } { f b, c + max f b, d }] /, ;. If s =, then b ad c / 8 { M a, c, d + M b, c, d ] / + M a, c, d + M b, c, d ] / }. Theorem 5 Let f : =a, b]c, d] R R have partial derivatives and f L. If f is an s, QC-convex function on the co-ordinates on with a < b and c < d for some s, ], >, and l, then. When s, ], b ad c / 6 l s l + s + s + { s + M a, c, d + s+ s 3M b, c, d ] /. When s =, where M u, c, d is defined by 9. b ad c 6 + s+ s 3M a, c, d + s + M b, c, d ] / } ; l / 4 /{M a, c, d l + / + ln M b, c, d ] / + ln M a, c, d + M b, c, d ] / }, Proof If s, ], similar to the proof of the ineuality 7, we can acuire

9 Wu and Qi SpringerPlus 65:49 Page 9 of 3 f ; a, b, c, d b ad c / / + + / / / / / / / + / / / { / / tλ l / tλ l/ t λ l/ / t λ l tλ l/ / tλ l t λ l/ / ] / ] / ] / t λ l / / l / ] / = b ad c 8 l { / / tλ l ] / / + t λ l / / + tλ l / + t λ l / / / 5 l 4/ b ad c l { s+l+3 s + s + l + ] / } ] / ] / ] / } s + M a, c, d + s+ s 3M b, c, d] / + s+ s+l+3 s 3M a, c, d + s + M b, c, d ]] / } s + s + l + b ad c / / = 6 l s l + s + s + { s + M a, c, d + s+ s 3M b, c, d ] / + s+ s 3M a, c, d + s + M b, c, d ] / }. If s =, similarly one can see that

10 Wu and Qi SpringerPlus 65:49 Page of 3 b ad c l { / / / 5 l 4/ ] / tλ l / + t λ l / / + tλ l / + t λ l / / / 5 l 4/ b ad c l { M a, c, d + ln M b, c, d ]] / l+ l + + ln M l+ a, c, d + M b, c, d ]] / } l + b ad c / 4 / = 6 l l + { M a, c, d + ln M b, c, d ] / + ln M a, c, d + M b, c, d ] / }. ] / ] / ] / } The proof of Theorem 5 is complete. Corollary 3 Under the conditions of Theorem 5, when l =,. If s, ], then b ad c / 3 s s + s + { s + M a, c, d + s+ s 3M b, c, d ] /. if s =, then Corollary 4 Under the conditions of Theorem 5, when l =,. If s, ], then + s+ s 3M a, c, d + s + M b, c, d ] / } ; b ad c/ { 3 M a, c, d + ln M b, c, d ] / + ln M a, c, d + M b, c, d ] / }.

11 Wu and Qi SpringerPlus 65:49 Page of 3 b ad c / 6 s + s + s + { s + M a, c, d + s+ s 3M b, c, d ] / + s+ s 3M a, c, d + s + M b, c, d ] } / ;. If s =, then b ad c 4 / 6 + { M a, c, d + ln M b, c, d ] / + ln M a, c, d + M b, c, d ] } /. Theorem 6 Let f : =a, b]c, d] R R have partial derivatives and f L. If f is an s, QC-convex function on the co-ordinates on with a < b and c < d for some s, ] and >, then b ad c / 4 s + { f a, c max, f a, d } { f b, c + max, f b, d }] /. / Proof From Lemma, Hölder s integral ineuality, the co-ordinated s, QC-convexity of f, and Lemma, it follows that f ; a, b, c, d b ad c / K t, λ / ] / / ] / b ad c 4 { { t s f a, c max, f a, d } { + t s f b, c max, f b, d }] } / dt b ad c / / = 4 s + { f a, c max, f a, d } { f b, c + max, f b, d }] /. Theorem 6 is thus proved.

12 Wu and Qi SpringerPlus 65:49 Page of 3 Conclusions Our main results in this paper are Definitions to and those integral ineualities of Hermite Hadamard type in Theorems 4 to 6. Authors contributions Both authors contributed eually to the manuscript. Both authors read and approved the final manuscript. Author details College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City 843, Inner Mongolia Autonomous Region, China. Department of Mathematics, School of Science, Tianjin Polytechnic University, Tianjin City 36, China. 3 Institute of Mathematics, Henan Polytechnic University, Jiaozuo City 454, Henan Province, China. Acknowledgements The authors thank the anonymous referees for their careful corrections to and valuable comments on the original version of this paper. This work was partially supported by the National Natural Science Foundation of China under Grant No and by the Inner Mongolia Autonomous Region Natural Science Foundation Project under Grant No. 5MS3, China. Competing interests The authors declare that they have no competing interests. Received: 8 October 5 Accepted: 6 January 6 References Alomari M, Darus M 8 Hadamard-type ineualities for s-convex functions. Int Math Forum 43: Bai S-P, Qi F, Wang S-H 6 Some new integral ineualities of Hermite C Hadamard type for α, m; P-convex functions on co-ordinates. J Appl Anal Comput 6:7 78. doi:.948/64 Dragomir SS On the Hadamard s ineuality for convex functions on the co-ordinates in a rectangle from the plane. Taiwan J Math 54: Dragomir SS, Pearce CEM 998 Quasi-convex functions and Hadamard s ineuality. Bull Aust Math Soc 573: doi:.7/s Dragomir SS, Pearce CEM Selected topics on Hermite Hadamard type ineualities and applications, RGMIA monographs. Victoria University, Melbourne, Australia. Hudzik H, Maligranda L 994 Some remarks on s-convex functions. Aeu Math 48:. doi:.7/bf83798 Hwang D-Y, Tseng K-L, Yang G-S 7 Some Hadamard s ineualities for co-ordinated convex functions in a rectangle from the plane. Taiwan J Math :63 73 Latif MA, Dragomir SS On some new ineualities for differentiable co-ordinated convex functions. J Ineual Appl :8. doi:.86/9-4x--8 Özdemir ME, Set E, Sarikaya MZ Some new Hadamard-type ineualities for co-ordinated m-convex and α, m -convex functions. Hacet J Math Stat 4:9 9 Özdemir ME, Akdemir AO, Yildiz Ç a On co-ordinated uasi-convex functions. Czechoslov Math J 64: doi:.7/s z Özdemir ME, Latif MA, Akdemir AO b On some Hadamard-type ineualities for product of two s-convex functions on the co-ordinates. J Ineual Appl :. doi:.86/9-4x-- Özdemir ME, Yildiz Ç, Akdemir AO c On some new Hadamard-type ineualities for co-ordinated uasi-convex functions. Hacettepe J Math Stat 45: Özdemir ME, Kavurmaci H, Akdemir AO, Avci M d Ineualities for convex and s-convex functions on =a, b]c, d]. J Ineual Appl :. doi:.86/9-4x-- Özdemir ME, Akdemir AO, Kavurmaci H 4 On the Simpson s ineuality for co-ordinated convex functions. Turkish J Anal Number Theory 5: doi:.69/tjant--5- Özdemir ME, Yildiz C, Akdemir AO 4 On the co-ordinated convex functions. Appl Math Inf Sci 83:85 9. doi:.785/amis/838 Pečarić J, Proschan F, Tong YL 99 Convex functions, partial orderings, and statistical applications. Mathematics in science and engineering, vol 87. Academic Press, New York Qi F, Xi B-Y 3 Some integral ineualities of Simpson type for GA-ε-convex functions. Georgian Math J 4: doi:.55/gmj-3-43 Roberts AW, Varberg DE 973 Convex functions. Academic Press, New York Sarikaya MZ, Set E, Özdemir ME, Dragomir SS New some Hadamard s type ineualities for co-ordinated convex functions. Tamsui Oxf J Math Sci 8:37 5 Wu Y, Qi F, Pei Z-L, Bai S-P 6 Hermite Hadamard type integral ineualities via s, m P-convexity on co-ordinates. J Nonlinear Sci Appl 93: Xi B-Y, Qi F Some integral ineualities of Hermite Hadamard type for convex functions with applications to means. J Funct Spaces Appl 98438:4. doi:.55//98438 Xi B-Y, Bai R-F, Qi F Hermite Hadamard type ineualities for the m- and α, m-geometrically convex functions. Aeu Math 843:6 69. doi:.7/s--4-x

13 Wu and Qi SpringerPlus 65:49 Page 3 of 3 Xi B-Y, Qi F 3 Some Hermite Hadamard type ineualities for differentiable convex functions and applications. Hacet J Math Stat 43:43 57 Xi B-Y, Qi F 5a Ineualities of Hermite Hadamard type for extended s-convex functions and applications to means. J Nonlinear Convex Anal 65: Xi B-Y, Qi F 5b Integral ineualities of Hermite Hadamard type for α, m, log-convex functions on co-ordinates. Probl Anal Issues Anal 4:7 9. doi:.5393/j3.art.5.89 Xi B-Y, Qi F 5c Some new integral ineualities of Hermite-Hadamard type for log, α, m-convex functions on coordinates. Stud Univ Babeş-Bolyai Math 64:59 55 Xi B-Y, Bai S-P, Qi F 5 Some new ineualities of Hermite Hadamard type for α, m -s, m -convex functions on coordinates. ResearchGate dataset. doi:.34/