q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions

Tam metin

(1)Journal of King Saud University – Science (2016) xxx, xxx–xxx. King Saud University. Journal of King Saud University – Science www.ksu.edu.sa www.sciencedirect.com. q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions _ _ ßcan c Necmettin Alp a, Mehmet Zeki Sarıkaya a, Mehmet Kunt b,*, Imdat Is a. Department of Mathematics, Faculty of Science and Arts, Du¨zce University, Du¨zce, Turkey Department of Mathematics, Faculty of Sciences, Karadeniz Technical University, 61080 Trabzon, Turkey c Department of Mathematics, Faculty of Sciences and Arts, Giresun University, 28200 Giresun, Turkey b. Received 4 September 2016; accepted 26 September 2016. KEYWORDS Hermite–Hadamard inequality; Midpoint type inequality; q-Integral inequalities; q-Derivative; q-Integration; Convexity; Quasi-convexity. Abstract In this paper, we prove the correct q-Hermite–Hadamard inequality, some new qHermite–Hadamard inequalities, and generalized q-Hermite–Hadamard inequality. By using the left hand part of the correct q-Hermite–Hadamard inequality, we have a new equality. Finally using the new equality, we give some q-midpoint type integral inequalities through q-differentiable convex and q-differentiable quasi-convex functions. Many results given in this paper provide extensions of others given in previous works. Ó 2016 The Authors. Production and hosting by Elsevier B.V. on behalf of King Saud University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).. 1. Introduction The study of calculus without limits is known as quantum calculus or q-calculus. The famous mathematician Euler initiated the study q-calculus in the eighteenth century by introducing the parameter q in Newton’s work of infinite series. In early twentieth century, Jackson (1910) has started a symmetric study of q-calculus and introduced q-definite integrals. The subject of quantum calculus has numerous applications in var* Corresponding author at: Department of Mathematics, Faculty of Sciences, Karadeniz Technical University, 61080 Trabzon, Turkey. E-mail address: [email protected] (M. Kunt). Peer review under responsibility of King Saud University.. Production and hosting by Elsevier. ious areas of mathematics and physics such as number theory, combinatorics, orthogonal polynomials, basic hyper-geometric functions, quantum theory, mechanics and in theory of relativity. This subject has received outstanding attention by many researchers and hence it is considered as an in-corporative subject between mathematics and physics. Interested readers are referred to Ernst (2012), Gauchman (2004), and Kac and Cheung (2001) for some current advances in the theory of quantum calculus and theory of inequalities in quantum calculus. In recent articles, Tariboon and Ntouyas (2013, 2014) studied the concept of q-derivatives and q-integrals over the intervals of the form ½a; b R and settled a number of quantum analogs of some well-known results such as Holder inequality, Hermite–Hadamard inequality and Ostrowski inequality, Cauchy–Bunyakovsky–Schwarz, Gruss, Gruss–Cebysev and other integral inequalities using classical convexity. Also, Noor et al. (2015), Noor et al. (2015),. http://dx.doi.org/10.1016/j.jksus.2016.09.007 1018-3647 Ó 2016 The Authors. Production and hosting by Elsevier B.V. on behalf of King Saud University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Alp, N. et al., q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions. Journal of King Saud University – Science (2016), http://dx.doi.org/10.1016/j.jksus.2016.09.007.

(2) 2. N. Alp et al.. Sudsutad et al. (2015), and Zhuang et al., 2016, have contributed to the ongoing research and have developed some integral inequalities which provide quantum estimates for the right part of the quantum analog of Hermite– Hadamard inequality through q-differentiable convex and q-differentiable quasi-convex functions. Let real function f be defined on some non-empty interval I of real line R. The function f said to be convex on I, if the inequality. Z. fðtÞ a dq t ¼ ð1 qÞðx aÞ a. fðta þ ð1 tÞbÞ 6 sup ffðaÞ; fðbÞg holds for all a; b 2 I and t 2 ½0; 1. Kırmacı (2004) obtained inequalities for differentiable convex mappings which are connected with midpoint type inequality, Alomari et al. (2009) obtained inequalities for differentiable quasi-convex mappings which are connected with midpoint type inequality. They used the following lemma to prove their theorems. . Lemma 1 Kırmacı (2004). Let f : I R ! R be a differentiable mapping on I ; a; b 2 I with a < b. If f 0 2 L½a; b, then the following equality holds: b 1 aþb fðtÞdt f ba a 2 "Z 1 # Z 1 2 ¼ba tf0 ðta þ ð1 tÞbÞdt þ ðt 1Þf 0 ðta þ ð1 tÞbÞdt Z. 1 2. ð1:1Þ. Z. Throughout this paper, let a < b and 0 < q < 1 be a constant. The following definitions and theorems for q-derivative and q-integral of a function f on ½a; b are given in Tariboon and Ntouyas (2013, 2014). Definition 2. For a continuous function f : ½a; b ! R then q-derivative of f at x 2 ½a; b is characterized by the expression fðxÞ fðqx þ ð1 qÞaÞ ; a Dq fðxÞ ¼ ð1 qÞðx aÞ. x – a:. ð2:1Þ. Since f : ½a; b ! R is a continuous function, thus we have ¼ limx!a a Dq fðxÞ. The function f is said to be q-differentiable on ½a; b if a Dq fðtÞ exists for all x 2 ½a; b. If a ¼ 0 in (2.1), then 0 Dq fðxÞ ¼ Dq fðxÞ, where Dq fðxÞ is familiar q-derivative of f at x 2 ½a; b defined by the expression (see Kac and Cheung, 2001) a Dq fðaÞ. Dq fðxÞ ¼. fðxÞ fðqxÞ ; ð1 qÞx. x – 0:. Z. x. ð2:2Þ. x. fðtÞ 0 dq t ¼. 1 X fðtÞdq t ¼ ð1 qÞx qn fðqn xÞ:. 0. ð2:4Þ. n¼0. If c 2 ða; xÞ, then the q-definite integral on ½c; x is expressed as Z. Z. x. c. Z. x. fðtÞ a dq t ¼. fðtÞa dq t a. c. fðtÞ a dq t:. ð2:5Þ. a. Theorem 4 Tariboon and Ntouyas (2014, Theorem 3.2). Let f : ½a; b ! R be a convex continuous function on ½a; b and 0 < q < 1. Then we have Z b aþb 1 qfðaÞ þ fðbÞ 6 : f fðtÞ a dq t 6 2 ba a 1þq. ð2:6Þ. _ ßcan (2016) give the following example to prove Kunt and Is that the left hand side of (2.6) is not correct: Example 5. Let ½a; b ¼ ½0; 1. Then the function fðtÞ ¼ 1 t is a convex continuous function on ½0; 1. Therefore the function f satisfies Theorem 4 assumptions. Then, from the inequality (2.6) the following inequality must be hold for all q 2 ð0; 1Þ f. 2. Preliminaries and definitions of q-calculus. ð2:3Þ. n¼0. 0. holds for all a; b 2 I and t 2 ½0; 1. The function f said to be quasi-convex on I, if the inequality. 1 X qn fðqn x þ ð1 qn ÞaÞ. for x 2 ½a; b. Rx Rx If a ¼ 0 in (2.3), then 0 fðtÞ 0 dq t ¼ 0 fðtÞdq t, where Rx 0 fðtÞdq t is familiar q-definite integral on ½0; x defined by the expression (see Kac and Cheung, 2001). fðta þ ð1 tÞbÞ 6 tfðaÞ þ ð1 tÞfðbÞ. 0. x. Z 1 0þ1 1 6 fðtÞ 0 dq t 2 10 0. 1 X 1 6 ð1 qÞ qn ð1 qn Þ 2 n¼0 1 1 1 6 ð1 qÞ 2 1 q 1 q2. 1. Then we have 1 q 6 : 2 1þq. ð2:7Þ. If we choose q ¼ 12 in (2.7) we have the following contradiction 1 1 6 : 2 3 It means that the left hand side of (2.6) is not correct. In the next section we give the correct q-Hermite– Hadamard inequality, some q-Hermite–Hadamard inequalities, and generalized q-Hermite–Hadamard inequality. 3. q-Hermite–Hadamard inequalities. Definition 3. Let f : ½a; b ! R be a continuous function. Then the q-definite integral on ½a; b is delineated as. In this section we prove q-Hermite–Hadamard inequality and varieties of q-Hermite–Hadamard inequalities.. Please cite this article in press as: Alp, N. et al., q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions. Journal of King Saud University – Science (2016), http://dx.doi.org/10.1016/j.jksus.2016.09.007.

(3) Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities Theorem 6 (q-Hermite–Hadamard inequality). Let f : ½a; b ! R be a convex differentiable function on ða; bÞ and 0 < q < 1. Then we have Z b qa þ b 1 qfðaÞ þ fðbÞ 6 : f fðxÞ a dq x 6 1þq ba a 1þq. Z. On the other hand, line connecting the points ða; fðaÞÞ and ðb; fðbÞÞ can be expressed as a function kðxÞ ¼ fðaÞþ fðbÞfðaÞ ba ðx aÞ. Since f is a convex function on ½a; b, than we have the following inequality. ð3:1Þ fðxÞ 6 kðxÞ ¼ fðaÞ þ. Proof. Since f is differentiable function on ða; bÞ, there is a 2 ða; bÞ. This tangent line for the function f at the point qaþb 1þq. 3. fðbÞ fðaÞ ðx aÞ ba. for all x 2 ½a; b (see Fig. 1). q-Integrating the inequality (3.4) on ½a; b, we have. fðbÞ fðaÞ fðaÞ þ ðx aÞ a dq x ba a Z fðbÞ fðaÞ b ¼ ðb aÞfðaÞ þ ðx aÞ a dq x ba a Z b fðbÞ fðaÞ x a dq x aðb aÞ ¼ ðb aÞfðaÞ þ ba a ! 1 X fðbÞ fðaÞ n n n ¼ ðb aÞfðaÞ þ ð1 qÞðb aÞ q ðð1 q Þa þ q bÞ aðb aÞ ba n¼0 fðbÞ fðaÞ 1 1 1 ð1 qÞðb aÞ aþ b aðb aÞ ¼ ðb aÞfðaÞ þ ba 1 q 1 q2 1 q2 qa þ b a ¼ ðb aÞfðaÞ þ ðfðbÞ fðaÞÞ 1þq fðbÞ fðaÞ ¼ ðb aÞfðaÞ þ ðb aÞ 1þq Z b qfðaÞ þ fðbÞ P ¼ ðb aÞ fðxÞ a dq x: 1þq a Z. b. ð3:4Þ. b. kðxÞ a dq x ¼ a. þ tangent line can be expressed as a function hðxÞ ¼ f qaþb 1þq x qaþb . Since f is a convex function on ½a; b, than f 0 qaþb 1þq 1þq we have the following inequality hðxÞ ¼ f. qa þ b qa þ b qa þ b þf0 x 6 fðxÞ 1þq 1þq 1þq. ð3:2Þ. for all x 2 ½a; b (see Fig. 1). q-Integrating the inequality (3.2) on ½a; b, we have. Z. b. A combination of (3.3) and (3.5) gives (3.1). Thus the proof is accomplished. h Remark 7. In Theorem 6, if we take q ! 1 , we recapture the well known Hermite–Hadamard inequality for convex function. Theorem 8. Let f : ½a; b ! R be a convex differentiable function on ða; bÞ and 0 < q < 1. Then we have. qa þ b qa þ b qa þ b þ f0 x f a dq x 1þq 1þq 1þq a Z b qa þ b qa þ b 0 qa þ b ¼ ðb aÞf x a dq x ðb aÞ þf 1þq 1þq 1þq a ! 1 X qa þ b qa þ b qa þ b þ f0 ð1 qÞðb aÞ qn ðð1 qn Þa þ qn bÞ ðb aÞ ¼ ðb aÞf 1þq 1þq 1þq n¼0 qa þ b qa þ b 1 1 1 qa þ b ¼ ðb aÞf a þ b ð b a Þ þ f0 ð1 qÞðb aÞ 1þq 1þq 1 q 1 q2 1 q2 1þq qa þ b qa þ b qa þ b qa þ b þ f0 ðb aÞ ðb aÞ ¼ ðb aÞf 1þq 1þq 1þq 1þq Z b qa þ b 6 fðxÞ a dq x: ¼ ðb aÞf 1þq a Z. ð3:5Þ. b. hðxÞ a dq x ¼ a. ð3:3Þ. Please cite this article in press as: Alp, N. et al., q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions. Journal of King Saud University – Science (2016), http://dx.doi.org/10.1016/j.jksus.2016.09.007.

(4) 4. N. Alp et al.. Fig. 1. Tangent and chord line for a convex function.. a þ qb ð1 qÞðb aÞ 0 a þ qb þ f f 1þq 1þq 1þq Z b 1 qfðaÞ þ fðbÞ : 6 fðxÞ a dq x 6 ba a 1þq. A combination of (3.5) and (3.8) gives (3.6). Thus the proof is accomplished. h ð3:6Þ. Theorem 9. Let f : ½a; b ! R be a convex differentiable function on ða; bÞ and 0 < q < 1. Then we have. Proof. Since f is differentiable function on ða; bÞ, there is a tangent line for the function f at the point aþqb 2 ða; bÞ. This 1þq þ tangent line can be expressed as a function h1 ðxÞ ¼ f aþqb 1þq 0 aþqb aþqb f 1þq x 1þq . Since f is a convex function on ½a; b, than. aþb ð1 qÞðb aÞ 0 a þ b f þ f 2 2ð1 þ qÞ 2 Z b 1 qfðaÞ þ fðbÞ : 6 fðxÞ a dq x 6 ba a 1þq. we have the following inequality a þ qb a þ qb a þ qb h1 ðxÞ ¼ f þf0 x 6 fðxÞ 1þq 1þq 1þq. Proof. Since f is differentiable function on ða; bÞ, there is a tangent line for the function f at the point aþb 2 ða; bÞ. This tan2 þ gent line can be expressed as a function h2 ðxÞ ¼ f aþb 2 . aþb f 0 aþb x . Since f is a convex function on ½a; b, we have 2 2 the following inequality. ð3:7Þ. for all x 2 ½a; b (see Fig. 1). q-Integrating the inequality (3.7) on ½a; b, we have Z a. b. a þ qb a þ qb 0 a þ qb þf x h1 ðxÞ a dq x ¼ f a dq x 1þq 1þq 1þq a Z b a þ qb a þ qb a þ qb þf0 ¼ ðb aÞf x a dq x ðb aÞ 1þq 1þq 1þq a ! 1 X a þ qb a þ qb 0 a þ qb n n n þf ð1 qÞðb aÞ q ðð1 q Þa þ q bÞ ðb aÞ ¼ ðb aÞf 1þq 1þq 1þq n¼0 a þ qb a þ qb 1 1 1 a þ qb þf0 ð1 qÞðb aÞ ¼ ðb aÞf a þ b ð b a Þ 1þq 1þq 1 q 1 q2 1 q2 1þq a þ qb a þ qb qa þ b a þ qb þf0 ðb aÞ ðb aÞ ¼ ðb aÞf 1þq 1þq 1þq 1þq Z b a þ qb ð1 qÞðb aÞ2 0 a þ qb þ 6 f ¼ ðb aÞf fðxÞ a dq x: 1þq 1þq 1þq a Z. ð3:9Þ. b. ð3:8Þ. Please cite this article in press as: Alp, N. et al., q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions. Journal of King Saud University – Science (2016), http://dx.doi.org/10.1016/j.jksus.2016.09.007.

(5) Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities aþb aþb aþb þf0 x 6 fðxÞ: h2 ðxÞ ¼ f 2 2 2. ð3:10Þ. 5. q-midpoint type integral inequalities through q-differentiable convex and q-differentiable quasi-convex functions. We will use the following Lemma to prove our main results.. for all x 2 ½a; b (see Fig. 1). q-Integrating the inequality (3.10) on ½a; b, we have. Z. aþb aþb aþb f þf0 x a dq x 2 2 2 a Z b aþb aþb 0 aþb ¼ ðb aÞf x a dq x ðb aÞ þf 2 2 2 a ! 1 X aþb aþb aþb þf0 ð1 qÞðb aÞ qn ðð1 qn Þa þ qn bÞ ðb aÞ ¼ ðb aÞf 2 2 2 n¼0 aþb aþb 1 1 1 aþb þf0 ð1 qÞðb aÞ ¼ ðb aÞf a þ b ð b a Þ 2 2 1 q 1 q2 1 q2 2 aþb a þ b qa þ b a þ b þf0 ðb aÞ ðb aÞ ¼ ðb aÞf 2 2 1þq 2 Z b 2 aþb ð1 qÞðb aÞ 0 a þ b þ 6 f ¼ ðb aÞf fðxÞ a dq x: 2 2 2ð1 þ qÞ a Z. b. b. h2 ðxÞ a dq x ¼ a. A combination of (3.5) and (3.11) gives (3.9). Thus the proof is accomplished. h Theorem 10. [Generalized q-Hermite–Hadamard inequality] Let f : ½a; b ! R be a convex differentiable function on ða; bÞ and 0 < q < 1. Then we have max fI1 ; I2 ; I3 g 6. 1 ba. Z. b. fðxÞ a dq x 6 a. qfðaÞ þ fðbÞ : 1þq. ð3:12Þ. Lemma 11. Let f : ½a; b ! R be a q-differentiable function on ða; bÞ. If a Dq f is continuous and integrable on ½a; b, then the following identity holds: f. Z b qa þ b 1 fðxÞ a dq x 1þq ðb aÞ a "Z 1 1þq t a Dq fðtb þ ð1 tÞaÞ0 dq t ¼ qðb aÞ 0. # 1 t a Dq fðtb þ ð1 tÞaÞ0 dq t þ 1 q 1þq Z. where. qa þ b ; 1þq a þ qb ð1 qÞðb aÞ 0 a þ qb þ f ; I2 ¼ f 1þq 1þq 1þq aþb ð1 qÞðb aÞ 0 a þ b þ f : I3 ¼ f 2 2ð1 þ qÞ 2. I1 ¼ f. ð3:11Þ. 1. ð4:1Þ. Proof. Using (2.1), we have a Dq fðtb þ ð1 tÞaÞ ¼. Proof. A combination of (3.1), (3.6), and (3.9) gives (3.12). Thus the proof is accomplished. h 4. Midpoint type inequalities via q-calculus. ¼. fðtb þ ð1 tÞaÞ fðq½tb þ ð1 tÞa þ ð1 qÞaÞ ð1 qÞ½tb þ ð1 tÞa a fðtb þ ð1 tÞaÞ fðqtb þ ð1 qtÞaÞ : tð1 qÞðb aÞ ð4:2Þ. Calculating following integrals by using (2.3) and (4.2), we have. In this section we proved an equality for the q-analog of midpoint type inequality. By using this equality we have. qðb aÞ. 1 R 1þq. R1 . . . t a Dq fðtb þ ð1 tÞaÞ0 dq t þ 1 t þ ð1 tÞaÞ0 dq t 1þq 1 R 1 R1 R ¼ qðb aÞ 01þq t a Dq fðtb þ ð1 tÞaÞ 0 dq t þ 1 t 1q a Dq fðtb þ ð1 tÞaÞ 0 dq t 1q 01þq a Dq fðtb þ ð1 tÞaÞ0 dq t 1þq R 1 þ1q 01þq a Dq fðtb þ ð1 tÞaÞ0 dq t 0. 1 D fðtb q a q. Please cite this article in press as: Alp, N. et al., q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions. Journal of King Saud University – Science (2016), http://dx.doi.org/10.1016/j.jksus.2016.09.007.

(6) 6. N. Alp et al. "Z. 1. ¼ qðb aÞ. t a Dq fðtb þ ð1 tÞaÞ0 dq t 0. 2. 1 q. Z. 1 a Dq fðtb þ ð1 tÞaÞ0 dq t þ 0. 3. Z. 1 1þq. # a Dq fðtb þ ð1 tÞaÞ0 dq t. 0. 3 fðtb þ ð1 tÞaÞ fðqtb þ ð1 qtÞaÞ0 dq t R 6 7 6 7 1 fðtbþð1tÞaÞ 1 7¼6 7 1q fðqtbþðt1qtÞaÞ 0 dq t ¼ qðb aÞ6 0 t 4 5 4 5 1 1 R R fðqtbþð1qtÞaÞ 1þq fðtbþð1tÞaÞ 1 1 1þq fðtbþð1tÞaÞfðqtbþð1qtÞaÞ d t þ d t þq 0 0 q 0 q t t 1q 0 tð1qÞðbaÞ " ! ! 1 1 1 1 X X X X ¼ q qn fðqn b þ ð1 qn ÞaÞ qn f qnþ1 b þ 1 qnþ1 a fðqn b þ ð1 qn ÞaÞ f qnþ1 b þ 1 qnþ1 a R1. Þfðqtbþð1qtÞaÞ t fðtbþð1ttðÞa1q 0 dq t 0 ÞðbaÞ R 1 1 fðtbþð1tÞaÞfðqtbþð1qtÞaÞ q 0 0 dq t tð1qÞðbaÞ. n¼0. 2. 1 q. q 1q. R1 0. n¼0. n¼0. n¼0. X !# 1 1 nþ1 X qn qn q qnþ1 þ bþ 1 a bþ 1 a f f 1þq 1þq 1þq 1þq n¼0 n¼0 ! X Z b 1 1 1 qa þ b qa þ b 1 n n n ¼ q fðbÞ q fðq b þ ð1 q ÞaÞ ðfðbÞ fðaÞÞ þ f fðxÞ a dq x: 1 fðaÞ ¼ f q q 1þq 1þq ðb aÞ a n¼0. Thus the proof is accomplished.. h. Remark 12. In Lemma 11, if we take q ! 1 , we recapture Lemma 1.. Proof. Taking value on both sides of (4.1) and using. absolute. the fact that a Dq f is convex on ½a; b, then we have. We can now prove some quantum estimates of q-midpoint type integral inequalities by using convexity and quasiconvexity of the absolute values of the q-derivatives.. . Z b. qa þ b. 1. f fðxÞ a dq x. 1 þ q ðb aÞ a "Z 1 Z 1þq. t a Dq fðtb þ ð1 tÞaÞ 0 dq t þ 6 qðb aÞ. # . 1 t a Dq fðtb þ ð1 tÞaÞ 0 dq t 1 q 0 1þq 2 3 1.

(7). R 1þq. 0 t t a Dq fðbÞ þ ð1 tÞ a Dq fðaÞ 0 dq t 5 . 6 qðb aÞ4 R 1 .

(8) þ 1 1q t t a Dq fðbÞ þ ð1 tÞ a Dq fðaÞ 0 dq t 1þq 2 3. R 1. R 1. a Dq fðbÞ 1þq t2 0 dq t þ a Dq fðaÞ 1þq tð1 tÞ0 dq t 0 0 5 6 qðb aÞ4. R . R . a Dq fðbÞ 11 1 t t0 dq t þ a Dq fðaÞ 11 1 t ð1 tÞ 0 dq t q q 1þq 1þq 2 3 . R 1 R . a Dq fðbÞ 1þq t2 0 dq t þ 11 1 t t 0 dq t 0 q 6 7 1þq 1 7 6 qðb aÞ6 4. 5. R 1þq R 1 1. a Dq fðaÞ. tð1 tÞ0 dq t þ 1 t ð1 tÞ0 dq t 0. 1þq. 1. ð4:4Þ. q. Theorem 13. Let f : ½a; b ! R be a q-differentiable function on ða; bÞ; a Dq f be continuous and integrable on ½a; b and. 0 < q < 1. If a Dq f is convex on ½a; b, then the following qmidpoint type inequality holds:. . Z b. qa þ b. 1. f. fðxÞ d x a q. 1þq. ðb aÞ a ". 3 6 qðb aÞ a Dq fðbÞ. 3 ð1 þ qÞ ð1 þ q þ q2 Þ #. 1 þ 2q þ 2q2 þ a Dq fðaÞ. ð1 þ qÞ3 ð1 þ q þ q2 Þ ð4:3Þ. We evaluate the appearing definite q-integrals as follows Z. 1 1þq. t2 0 dq t ¼ ð1 qÞ. 0. n 2 1 1 X q qn 1þq 1 þ q n¼0. ð4:5Þ 1 1 ¼ ; 3 3 ð 1 þ qÞ 1 q ð1 þ qÞ ð1 þ q þ q2 Þ Z 1þq Z 1þq Z 1þq 1 1 1 tð1 tÞ0 dq t ¼ t 0 dq t t2 0 dq t 0 0 0 n 1 1 X q 1 ¼ ð 1 qÞ qn 3 1 þ q n¼0 1þq ð1 þ qÞ ð1 þ q þ q2 Þ 1 1 ¼ ð1 þ qÞ3 ð1 þ qÞ3 ð1 þ q þ q2 Þ q ; ¼ 2 ð 1 þ qÞ ð 1 þ q þ q2 Þ ¼ ð1 qÞ. 1. 3. ð4:6Þ. Please cite this article in press as: Alp, N. et al., q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions. Journal of King Saud University – Science (2016), http://dx.doi.org/10.1016/j.jksus.2016.09.007.

(9) Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities Z 1 Z 1þq Z 1 1 1 1 1 t t 0 dq t ¼ t t 0 dq t t t 0 dq t 1 q q q 0 0 1þq Z Z 1þq Z 1 Z 1þq 1 1 1 1 1 t 0 dq t t2 0 dq t t 0 dq t þ t2 0 d q t ¼ q 0 q 0 0 0 " 1 1 1 X X 1X 1 q2n q3n q2n ¼ ð1 qÞ 2 q n¼0 qð1 þ qÞ n¼0 n¼0 # 1 1 X 3n q þ ð1 þ qÞ3 n¼0 " 1 1 1 ¼ qð1 þ qÞ 1 þ q þ q2 qð1 þ qÞ3 # 1 þ ð1 þ qÞ3 ð1 þ q þ q2 Þ 2 ¼ ; ð4:7Þ ð1 þ qÞ3 ð1 þ q þ q2 Þ. Z. qa þ b 1. f. 1 þ q ðb a Þ. b. a. Z. fðxÞ a dq x. 6 qðb aÞ. Making use of (4.4)–(4.8), gives us the desired result (4.3). Thus the proof is accomplished. h Corollary 14. In Theorem 13, if we take q ! 1 , we have the following midpoint type inequality for convex functions:. . Z b. aþb. 1. f. fðxÞdx . ðb aÞ a 2 6. ðb aÞ½jf 0 ðaÞj þ jf 0 ðbÞj : 8. Z 1 Z 1 1 1 1 t ð1 tÞ 0 dq t ¼ t 0 dq t t t 0 dq t 1 1 1 q q q 1þq 1þq 1þq Z 1 Z 1þq 1 1 1 t 0 dq t t 0 dq t ¼ q q 0 0 2 ð1 þ qÞ3 ð1 þ q þ q2 Þ 1 þ q þ q2 : ¼ ð1 þ qÞ3 ð1 þ q þ q2 Þ ð4:8Þ. Theorem 16. Let f : ½a; b ! R be a q-differentiable function on ða; bÞ; a Dq f be continuous and integrable on ½a; b and. r 0 < q < 1. If a Dq f is convex on ½a; b for r P 1, then the following q-midpoint type inequality holds:. 1þq. ð4:10Þ. Proof. Taking absolute value on both sides of (4.1), applying. r the power mean inequality and using the fact that a Dq f is convex on ½a; b for r P 1, we get that Making use of (4.5)–(4.8) in (4.11), gives us the desired result (4.10). Thus the proof is accomplished. h. . Rb. qaþb. 1 fðxÞ d x. f 1þq ðba. a q Þ a 1 . R1 R 6 qðb aÞ 01þq t a Dq fðtb þ ð1 tÞaÞ 0 dq t þ 1 1q t a Dq fðtb þ ð1 tÞaÞ 0 dq t 1þq 2 3 1 11r 1 1r. r R 1þq. R 1þq. 6 7 0 t 0 dq t 0 t a Dq fðtb þ ð1 tÞaÞ 0 dq t 6 7 6 qðb aÞ6 11r R . 1r 7 4 R 1 5. r 1 1 þ 1 1q t 0 dq t t a Dq fðtb þ ð1 tÞaÞ 0 dq t 1 q 1þq 1þq 2 3 1 1r. r. r

(10) R 1þq . 6 7 0 dq t 0 t t a Dq fðbÞ þ ð1 tÞ a Dq fðaÞ 6 7 6 qðb aÞ 1 33 6 7 r 1 ð1þqÞ 4 5. r. r

(11). R1 1 r. þ 1 t t a Dq fðbÞ þ ð1 tÞ a Dq fðaÞ 0 dq t 2. ð4:9Þ. Remark 15. In (4.9), we recapture the inequality Kırmacı, 2004, Theorem 2.2.. 2 1r 3. q 1. a Dq fðbÞ r. a Dq fðaÞ r þ 3 2 7 6 1 ð1þqÞ ð1þqþq2 Þ ð1þqÞ ð1þqþq2 Þ 7 6 1r 5: 33r 4 . 2 r r ð1 þ qÞ 1þqþq 2. þ a Dq fðbÞ ð1þqÞ3 ð1þqþq2 Þ þ a Dq fðaÞ ð1þqÞ3 ð1þqþq2 Þ. 1. 7. ð4:11Þ. q. 3 11r. R 1. a Dq fðbÞ r 1þq t2 0 dq t 0 6 7 @ A 6 7 1. r R 1þq 6 7. þ D fðaÞ t ð 1 t Þ d t a q 0 q 6 7 0 1 7: 1 0 6 qðb aÞ 1 33 6 . R r 7 ð1þqÞ r 6. a Dq fðbÞ r 11 1 t t 0 dq t 6 7 q 6 B C 7 1þq 4 þ@. A 5. r R 1 þ a Dq fðaÞ 1 1q t ð1 tÞ 0 dq t 0. 1þq. Please cite this article in press as: Alp, N. et al., q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions. Journal of King Saud University – Science (2016), http://dx.doi.org/10.1016/j.jksus.2016.09.007.

(12) 8. N. Alp et al.. Corollary 17. In Theorem 16, if we take q ! 1 , we have the following midpoint type inequality for convex functions:. . Z b. aþb. 1. f. fðxÞdx. 2 ðb aÞ a 2 3 r 1 r 1 0 0 1 4 jf ðbÞj 24 þ jf ðaÞj 121 r 5 6 ðb aÞ 33 1 : 2 r þ jf 0 ðbÞjr 1 þ jf 0 ðaÞjr 1 r 12 24. ð4:12Þ. Theorem 18. Let f : ½a; b ! R be a q-differentiable function on ða; bÞ; a Dq f be continuous and integrable on ½a; b and 0 < q < 1.. r If a Dq f is convex on ½a; b for r > 1, the following q-midpoint type inequality holds:. . Rb. qaþb. 1 fðxÞ a dq x. f 1þq ðba Þ a 2 3 1 r 1s D fðbÞ r ja q j ð2qþq2 Þja Dq fðaÞj r 1q 1 6 7 þ ð1þqÞsþ1 1qsþ1 ð1þqÞ3 ð1þqÞ3 6 7 7 6 qðb aÞ6 1r 7 6 1 r r s 4 R1 1 s ð2qþq2 Þja Dq fðbÞj ðqþq2 þq3 Þja Dq fðaÞj 5 þ 1 q t 0 dq t þ ð1þqÞ3 ð1þqÞ3 1þq. ð4:13Þ. where r. 1. þs. 1. ¼ 1.. Proof. Taking absolute value on both sides of (4.1), applying. r the Ho¨lder inequality and using the fact that a Dq f is convex on ½a; b for r > 1, we get that. Corollary 19. In Theorem 18, If we take q ! 1 , we have the following midpoint type inequality for convex functions:. . Z b. aþb. 1. f fðxÞdx. 2 ðb aÞ a 2 3 1s 0 r r 1 jf ðbÞj þ 3 jf 0 ðaÞj r ðb aÞ 4 4 5: 6 r r 1 16 sþ1 þ 3 jf 0 ðbÞj þ jf 0 ðaÞj r. ð4:14Þ. Remark 20. In (4.14), we recapture the inequality Kırmacı, 2004, Theorem 2.3. Theorem 21. Let f : ½a; b ! R be a q-differentiable function on ða; bÞ; a Dq f be continuous and integrable on ½a; b and 0 < q < 1.. r If a Dq f is convex on ½a; b for r > 1, the following q-midpoint type inequality holds:. . 3s Z b. qa þ b. 1 1. f. fðxÞ d x 6 q ð b a Þ a q. 1þq ðb aÞ a 1þq 2 . 1r 3. r. r q 1. a Dq fðbÞ. þ D fðaÞ a q 6 7 ð1þqÞ3 ð1þqþq2 Þ ð1þqÞ2 ð1þqþq2 Þ 7 6 4 . 1 5 . r. r 1þqþq2 r 2. þ a Dq fðbÞ ð1þqÞ3 ð1þqþq2 Þ þ a Dq fðaÞ ð1þqÞ3 ð1þqþq2 Þ ð4:15Þ where r1 þ s1 ¼ 1.. . Rb. qaþb. 1 fðxÞa dq x. f 1þq ðba Þ a 1 . R1 R 6 qðb aÞ 01þq t a Dq fðtb þ ð1 tÞaÞ 0 dq t þ 1 1q t a Dq fðtb þ ð1 tÞaÞ 0 dq t 1þq 2 3 1 1s 1 1r. r R 1þq. R 1þq s. a Dq fðtb þ ð1 tÞaÞ 0 dq t 6 7 0 t 0 dq t 0 6 7 6 qðb aÞ6 7 s 1s R. 1r 5 4 R . r 1 1 1. þ 1 q t 0 dq t 1 a Dq fðtb þ ð1 tÞaÞ 0 dq t 1þq 1þq 2 3 1 1r s 1 1 . r. r

(13) R 1þq R 1þq s. 6 t a Dq fðbÞ þ ð1 tÞ a Dq fðaÞ 0 dq t 7 0 t 0 dq t 0 6 7 6 qðb aÞ6 7 1 5 s 1s R . 4 R . r. r

(14) r 1 1. þ 1 1q t 0 dq t t D fðbÞ þ ð 1 t Þ D f ð a Þ d t 1 a q a q 0 q 1þq 1þq 2 3 1 0 1 1. R r 1 1s. a Dq fðbÞ r 1þq t 0 dq t R 1þq s 0 6 7 @ A t d t 6 7 0 q 0 1. r R 1þq 6 7. þ a Dq fðaÞ 0 ð1 tÞ 0 dq t 6 7 7 6 qðb aÞ6 1 6. r R 1 0 1r 7 6 7. s 1 1 t 0 dq t a Dq fðbÞ 6 7 R 1þq A 5 4 þ 11 1 t 0 dq t s @. R q r 1 1þq þ a Dq fðaÞ 1 ð1 tÞ0 dq t 1þq 2 3 1 1s D fðbÞ r r 2 2qþq ð Þja Dq fðaÞj r ja q j 1q 1 6 7 þ ð1þqÞsþ1 1qsþ1 ð1þqÞ3 ð1þqÞ3 6 7 7 6 qðb aÞ6 1r 7 6 1 r r s R1 1 4 s ð2qþq2 Þja Dq fðbÞj ðqþq2 þq3 Þja Dq fðaÞj 5 þ 1 q t 0 dq t þ ð1þqÞ3 ð1þqÞ3 1þq. Thus the proof is accomplished.. h. Please cite this article in press as: Alp, N. et al., q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions. Journal of King Saud University – Science (2016), http://dx.doi.org/10.1016/j.jksus.2016.09.007.

(15) Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities Proof. Taking absolute value on both sides of (4.1), applying. r the Ho¨lder inequality and using the fact that a Dq f is convex on ½a; b for r > 1, we get that. 9. Theorem 23. Let f : ½a; b ! R be a q-differentiable function on ða; bÞ; a Dq f be continuous and integrable on ½a; b and. r 0 < q < 1. If a Dq f is quasi-convex on ½a; b for r P 1, the following q-midpoint type inequality holds:. . Rb. qaþb. 1 fðxÞ d x. f 1þq ðba. a q Þ a 2 3 1. R 1þq 1 1 s t r D fðtb þ ð1 tÞaÞ d t t a q 0 q 0 6 7 6 qðb aÞ4 R 1s 1r. 0 dq t5 1 1 1. þ 1 q t q t a Dq fðtb þ ð1 tÞaÞ 1þq 2 3 1 1s 1 1r. r R 1þq. R 1þq. 6 7 0 t 0 dq t 0 t a Dq fðtb þ ð1 tÞaÞ 0 dq t 6 7 6 qðb aÞ6 7 1s R . 1r 5 4 R 1 . r 1 1 1. þ 1 q t 0 dq t t a Dq fðtb þ ð1 tÞaÞ 0 dq t 1 q 1þq 1þq 2 3 1 1r. r R 1þq. 7 3s 6 0 t a Dq fðtb þ ð1 tÞaÞ 0 dq t 6 7 1 6 qðb aÞ 1þq 6 7 . 1r 5 4 R 1 . r 1. þ 1 t a Dq fðtb þ ð1 tÞaÞ 0 dq t 2. 1þq. ð4:16Þ. q. 3. R 1. R 1. a Dq fðbÞ r 1þq t2 0 dq t þ a Dq fðaÞ r 1þq tð1 tÞ0 dq t 3s 0 0 1 4. 5: 6 qðb aÞ 1þq. R . R . a Dq fðbÞ r 11 1 t t 0 dq t þ a Dq fðaÞ r 11 1 t ð1 tÞ0 dq t q q 1þq. 1þq. Making use of (4.5), (4.8) in (4.16), gives us the desired result (4.15). Thus the proof is accomplished. h . Corollary 22. In Theorem 21, if we take q ! 1 , we have the following midpoint type inequality for convex functions:. . Rb. aþb. 1 fðxÞdx. f 2 ðba. Þ a 2 3 r r 1 1 3s jf 0 ðbÞj 241 þ jf 0 ðaÞj 121 r 5 6 ðb aÞ 2 4 r r 1 þ jf 0 ðbÞj 121 þ jf 0 ðaÞj 241 r. ð4:17Þ. Some results related to quasi-convexity are presented in the following theorems.. Z. qa þ b 1. f . 1þq ðb aÞ. a. 6 ðb aÞ. 2q ð1 þ qÞ3. b. fðxÞ a dq x. . sup a Dq fðaÞ ; a Dq fðbÞ :. ð4:18Þ. Proof. Taking absolute value on both sides of (4.1), applying. r the power mean inequality and using the fact that a Dq f is quasi-convex on ½a; b for r P 1, we get that Hence the inequality (4.18) is established. Thus the proof is accomplished. h. . Rb. qaþb. 1 fðxÞ d x. f 1þq ðba. a q Þ a 1 . R1 R 6 qðb aÞ 01þq t a Dq fðtb þ ð1 tÞaÞ 0 dq t þ 1 1q t a Dq fðtb þ ð1 tÞaÞ 0 dq t 1þq 2 3 1 11r 1 1r. r R 1þq. R 1þq. 6 7 t d t t D f ð tb þ ð 1 t Þa Þ d t 0 q a q 0 q 0 0 6 7 6 qðb aÞ6 7 11r R . 1r 5 4 R 1 . r 1 1 1. þ 1 q t 0 dq t t a Dq fðtb þ ð1 tÞaÞ 0 dq t 1 q 1þq 1þq 2 3 1 11r 1 1r. r. r

(16). R 1þq R 1þq. a Dq fðaÞ ; a Dq fðbÞ 0 dq t 6 7 0 t 0 dq t 0 t sup 6 7 6 qðb aÞ6 7 11r R 1r 5 4 R 1 .

(17) r r 1 1 1. þ 1 q t 0 dq t t sup a Dq fðaÞ ; a Dq fðbÞ 1 0 dq t q 1þq 1þq 1 . r. r

(18) 1r R 1þq. R 1 1 t 0 dq t 6 qðb aÞ sup a Dq fðaÞ ; a Dq fðbÞ. 1 0 t 0 dq t þ 1þq q. 2q sup a Dq fðaÞ ; a Dq fðbÞ : 6 ðb aÞ ð1þq Þ3. Please cite this article in press as: Alp, N. et al., q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions. Journal of King Saud University – Science (2016), http://dx.doi.org/10.1016/j.jksus.2016.09.007.

(19) 10. N. Alp et al.. . Z b. aþb. 1. f. fðxÞdx. 2 ðb aÞ a 1 s ðb aÞ 1 sup fjf 0 ðaÞj; jf 0 ðbÞjg: 6 2 sþ1. Corollary 24. In Theorem 23, if we take q ! 1 , we have the following midpoint type inequality for quasi-convex functions:. Z. aþb 1. f . 2 ðb aÞ. a. 6. b. fðxÞdx. ðb aÞ sup fjf 0 ðaÞj; jf 0 ðbÞjg 4. ð4:19Þ. ð4:20Þ. Competing interests The authors declare that they have no competing interests.. Theorem 25. Let f : ½a; b ! R be a q-differentiable function on ða; bÞ; a Dq f be continuous and integrable on ½a; b and 0 < q < 1.. r If a Dq f is quasi-convex on ½a; b for r > 1, the following qmidpoint type inequality holds:. Acknowledgements The authors are very grateful to the referees for helpful comments and valuable suggestions.. . Z b. qa þ b. 1. f. fðxÞ d x a q. 1þq. ðb aÞ a. . 6 qðb aÞ sup a Dq fðaÞ ; a Dq fðbÞ. 2 !1s 1r 1 1 q 1 4 1þq ð1 þ qÞsþ1 1 qsþ1 3 1 ! s 1r Z 1 s 1 q 5 t 0 dq t þ 1 q 1 þ q 1þq. References Alomari, M., Darus, M., Dragomir, S.S., 2009. New inequalities of Hermite–Hadamard type for functions whose second derivatives absolute values are quasi-convex. RGMIA Res. Rep. Coll., 1–5 Article 14, 12 Supplement. Ernst, T., 2012. A Comprehensive Treatment of q-calculus. Springer, Basel. Gauchman, H., 2004. Integral Inequalities in q-calculus. Comput. Math. Appl. 47, 281–300. Jackson, F.H., 1910. On a q-definite integrals. Q. J. Pure Appl. Math. 41, 193–203. Kac, V., Cheung, P., 2001. Quantum Calculus. Springer, New York. Kırmacı, U.S., 2004. Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula. Appl. Math. Comput. 147, 137–146.. where r1 þ s1 ¼ 1. Proof. Taking absolute value on both sides of (4.1), applying. r the Ho¨lder inequality and using the fact that a Dq f is quasiconvex on ½a; b for r > 1, we get that. . Rb. qaþb. 1 fðxÞ a dq x. f 1þq ðba Þ a 1 . R1 R 6 qðb aÞ 01þq t a Dq fðtb þ ð1 tÞaÞ 0 dq t þ 1 1q t a Dq fðtb þ ð1 tÞaÞ 0 dq t 1þq 2 3 1 1s 1 1r. r R 1þq. R 1þq s. 6 7 a Dq fðtb þ ð1 tÞaÞ 0 dq t 0 t 0 dq t 0 6 7 6 qðb aÞ6 7 s 1s R. 1r 5 4 R . r 1 1. þ 1 1q t 0 dq t D f ð tb þ ð 1 t Þa Þ d t 1 0 q a q 1þq 1þq 2 1r 3 1s R 1 . r. r

(20). 1q 1þq 1. 6 sup a Dq fðaÞ ; a Dq fðbÞ 7 0 dq t 0 ð1þqÞsþ1 1qsþ1 6 7 6 qðb aÞ6 7 s 1s R 1r 5 4 R .

(21) r r 1 1 1 þ 1 t 0 dq t sup a Dq fðaÞ ; a Dq fðbÞ 0 dq t 1 1þq. q. 1þq. r. r

(22) 1. 6 qðb aÞ sup a Dq fðaÞ ; a Dq fðbÞ r " 1r 1s R 1 s 1s R R 1 1 1 1þq 1q d t ð1þq1Þsþ1 1q þ t d t 1 1 0 q 0 q sþ1 0 q 1þq. 1þq. 0 dq t. 1r. #. . 6 qðb aÞ sup a Dq fðaÞ ; a Dq fðbÞ. 1s 1r R s 1s 1r 1 1q q 1 1 1 þ t 0 dq t 1þq ð1þqÞsþ1 1qsþ1 1 1þq q 1þq. Thus the proof is accomplished.. h. Corollary 26. In Theorem 25, if we take q ! 1 , we have the following midpoint type inequality for quasi-convex functions:. _ ßcan, I., _ 2016. Erratum: Quantum integral inequalities on Kunt, M., Is finite intervals, doi: http://dx.doi.org/10.13140/RG.2.1.5059.4806, https://www.researchgate.net/publication/305303595, 1–2.. Please cite this article in press as: Alp, N. et al., q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions. Journal of King Saud University – Science (2016), http://dx.doi.org/10.1016/j.jksus.2016.09.007.

(23) Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities Noor, M.A., Noor, K.I., Awan, M.U., 2015. Some quantum estimates for Hermite–Hadamard inequalities. Appl. Math. Comput. 251, 675–679. Noor, M.A., Noor, K.I., Awan, M.U., 2015. Some quantum integral inequalities via preinvex functions. Appl. Math. Comput. 269, 242– 251. Sudsutad, W., Ntouyas, S.K., Tariboon, J., 2015. Quantum integral inequalities for convex functions. J. Math. Inequal. 9 (3), 781–793.. 11. Tariboon, J., Ntouyas, S.K., 2013. Quantum calculus on finite intervals and applications to impulsive difference equations. Adv. Diff. Equ. 282, 1–19. Tariboon, J., Ntouyas, S.K., 2014. Quantum integral inequalities on finite intervals. J. Inequal. Appl. 121, 1–13. Zhuang, H., Liu, W., Park, J., 2016. Some quantum estimates of Hermite-Hadamard inequalities for quasi-convex functions. Miskolc Math. Notes. 17 (2).. Please cite this article in press as: Alp, N. et al., q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions. Journal of King Saud University – Science (2016), http://dx.doi.org/10.1016/j.jksus.2016.09.007.

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