International Journal of Analysis and Applications ISSN 2291-8639 Volume 13, Number 2 (2017), 119-131 http://www.etamaths.com NEW INEQUALITIES OF HERMITE-HADAMARD TYPE FOR n-TIMES DIFFERENTIABLE s-CONVEX FUNCTIONS WITH APPLICATIONS MUHAMMAD AMER LATIF1,∗, SEVER S. DRAGOMIR2 AND EBRAHIM MOMONIAT1 Abstract. In this paper, some new inequalities Hermite-Hadamard type are obtained for functions whose nth derivatives in absolute value are s-convex functions. From our results, several inequalities of Hermite-Hadamard type can be derived in terms of functions whose first and second derivatives in absolute value are s-convex functions as special cases. Our results may provide refinements of some results already exist in literature. Applications to trapezoidal rule and to special means of established results are given. 1. Introduction A function f : I → R, ∅ 6= I ⊆ R, is said to be convex on I if the inequality f(tx + (1 − t)y) ≤ tf(x) + (1 − t)f(y) holds for all x,y ∈ I and t ∈ [0, 1]. Let f : I ⊆ R → R be a convex mapping and a,b ∈ I with a < b. Then f ( a + b 2 ) ≤ 1 b−a ∫ b a f(x)dx ≤ f(a) + f(b) 2 . (1.1) The double inequality (1.1) is known as the Hermite-Hadamard inequality (see [4]). The inequalities (1.1) hold in reversed direction if f is concave. For recent results, refinements, counterparts, generalizations and new Hermite-Hadamard-type in- equalities see [3, 7, 10–12, 14] and the references therein. In [4], Hudzik and Maligranda considered among others the class of functions which are s-convex in the second sense and is defined as follows. Definition 1.1. [4] Let s ∈ (0, 1] be a fixed real number. A function f : [0,∞) → [0,∞) is said to be s-convex in the second sense, if f (tx + (1 − t) y) ≤ tsf (x) + (1 − t)s f (y) (1.2) holds for all x, y ∈ [0,∞) and t ∈ [0, 1]. The class of s-convex functions in the second sense is denoted by K2s . If the inequality (1.2) holds in reversed direction, then f is to be an s-concave function in the second sense. It is clear that the definition of s-convexity (s-concavity) coincides with the definition of convexity (concavity) when s = 1. In [2], Dragomir and Fitzpatrick proved a variant of Hadamard’s inequality which holds for s-convex functions in the second sense. Theorem 1.1. [2] Suppose that f : [0,∞) → [0,∞) is an s-convex function in the second sense, where s ∈ (0, 1), a, b ∈ [0,∞) with a < b. If a, b ∈ L ([a,b]), then the following inequalities hold 2s−1f ( a + b 2 ) ≤ 1 b−a ∫ b a f(x)dx ≤ f(a) + f(b) s + 1 . (1.3) Received 8th November, 2016; accepted 11th January, 2017; published 1st March, 2017. 2010 Mathematics Subject Classification. Primary 26D07; Secondary 26D15. Key words and phrases. Hermite-Hadamard’s inequality; convex function; s-convex; Hölder integral inequality; trape- zoidal rule. c©2017 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 119 120 LATIF, DRAGOMIR AND MAMONAIT For more recent results on Hermite-Hadamard type inequalities for functions whose derivatives in absolute value are s-convex functions, we refer the interested reader to [1, 8, 9, 13] and the references therein. The main purpose of the present paper is to establish new Hermite-Hadamard type inequalities for functions whose nth derivatives in absolute value are s-convex. We believe that the results presented in this paper are better than those already exist in the literature concerning the inequalities of Hermite- Hadamard type for s-convex functions. Applications of our results to trapezoidal formula and to special means are given in Section 3 and Section 4. 2. Main Results We will use the following Lemmas to establish our main results in this section. Lemma 2.1. Let f : I ⊂ R → R be a function such that f(n) exists on I◦ and f(n) ∈ L ([a,b]), where a, b ∈ I◦ with a < b, n ∈ N, we have the identity f (a) + f (b) 2 − 1 b−a ∫ b a f (x) dx− n−1∑ k=1 k [ 1 + (−1)k ] (b−a)k 2k+1 (k + 1)! f(k) ( a + b 2 ) = (b−a)n 2n+1n! ∫ 1 0 (1 − t)n−1 (n− 1 + t) f(n) ( 1 − t 2 a + 1 + t 2 b ) dt + (−1)n (b−a)n 2n+1n! ∫ 1 0 (1 − t)n−1 (n− 1 + t) f(n) ( 1 − t 2 b + 1 + t 2 a ) dt, (2.1) where an empty sum is understood to be nil. Proof. Suppose In = (b−a)n 2n+1n! ∫ 1 0 (1 − t)n−1 (n− 1 + t) f(n) ( 1 − t 2 a + 1 + t 2 b ) dt and Jn = (−1)n (b−a)n 2n+1n! ∫ 1 0 (1 − t)n−1 (n− 1 + t) f(n) ( 1 − t 2 b + 1 + t 2 a ) dt. For n = 1, we have I1 = b−a 4 ∫ 1 0 tf ′ ( 1 − t 2 a + 1 + t 2 b ) dt and J1 = (−1) (b−a) 4 ∫ 1 0 tf ′ ( 1 − t 2 a + 1 + t 2 b ) dt. By integration by parts and using the substitution x = 1−t 2 a + 1+t 2 b for I1 and x = 1+t 2 a + 1−t 2 b for J1, we obtain I1 = 1 2 f (b) − 1 b−a ∫ b a+b 2 f (x) and J1 = 1 2 f (a) − 1 b−a ∫ a+b 2 a f (x) . Hence I1 + J1 = f (a) + f (b) 2 − 1 b−a ∫ b a f (x) dx, which coincides with the L.H.S of (2.1) for n = 1. Similarly for n = 2, and using similar arguments as above, we have I2 + J2 = f (a) + f (b) 2 − 1 b−a ∫ b a f (x) dx INEQUALITIES FOR n-TIMES DIFFERENTIABLE s-CONVEX FUNCTIONS 121 which coincides with the L.H.S of (2.1) for n = 2. Suppose (2.1) holds for n = m− 1 ≥ 3. Now for n = m, we have (b−a)m 2m+1m! ∫ 1 0 (1 − t)m−1 (m− 1 + t) f(m) ( 1 − t 2 a + 1 + t 2 b ) dt + (−1)m (b−a)m 2m+1m! ∫ 1 0 (1 − t)m−1 (m− 1 + t) f(m) ( 1 − t 2 b + 1 + t 2 a ) dt = − (b−a)m−1 (m− 1) [ 1 + (−1)m−1 ] 2mm! f(m−1) ( a + b 2 ) + (b−a)m−1 2m (m− 1)! ∫ 1 0 (1 − t)m−2 (m− 2 + t) f(m) ( 1 − t 2 a + 1 + t 2 b ) dt + (−1)m−1 (b−a)m−1 2m (m− 1)! ∫ 1 0 (1 − t)m−2 (m− 2 + t) f(m) ( 1 − t 2 b + 1 + t 2 a ) dt = f (a) + f (b) 2 − 1 b−a ∫ b a f (x) dx− m−2∑ k=1 k [ 1 + (−1)k ] (b−a)k 2k+1 (k + 1)! f(k) ( a + b 2 ) − (b−a)m−1 (m− 1) [ 1 + (−1)m−1 ] 2mm! f(m−1) ( a + b 2 ) = f (a) + f (b) 2 − 1 b−a ∫ b a f (x) dx− m−1∑ k=1 k [ 1 + (−1)k ] (b−a)k 2k+1 (k + 1)! f(k) ( a + b 2 ) . This completes the proof of the lemma. � Lemma 2.2. [15] Let x ≥ 0, y ≥ 0, the inequality (x + y) θ ≤ xθ + yθ holds for 0 < θ ≤ 1 and the inequality (x−y)θ ≤ xθ −yθ holds for θ ≥ 1. Now we state and prove some new Hermite-Hadamard type inequalities for functions whose nth derivatives in absolute value are s-convex and s-concave in the second sense. Theorem 2.1. Let f : I ⊂ [0,∞) → R be a function such that f(n) exists on I◦ and f(n) ∈ L ([a,b]), where a, b ∈ I◦ with a < b, n ∈ N. If ∣∣f(n)∣∣q is s-convex on [a,b] for some fixed s ∈ (0, 1] and q ∈ [1,∞), we have the inequality∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx − n−1∑ k=1 k [ 1 + (−1)k ] (b−a)k 2k+1 (k + 1)! f(k) ( a + b 2 )∣∣∣∣∣∣ ≤ (b−a) n 2n+s/q+1n! ( n n + 1 )1−1/q ×   (( n2 + s (n− 1) )∣∣f(n) (a)∣∣q (n + s) (n + s + 1) + ( n n + 1 + n (n + s) n + s + 1 B (s + 1,n) )∣∣∣f(n) (b)∣∣∣q )1/q + (( n2 + s (n− 1) )∣∣f(n) (b)∣∣q (n + s) (n + s + 1) + ( n n + 1 + n (n + s) n + s + 1 B (s + 1,n) )∣∣∣f(n) (a)∣∣∣q )1/q  , (2.2) 122 LATIF, DRAGOMIR AND MAMONAIT where B (α,β) = ∫ 1 0 tα−1 (1 − t)β−1 dt,α,β > 0 is the Euler Beta function. Proof. From Lemma 2.1, the Hölder inequality and s-convexity of ∣∣f(n)∣∣q on [a,b], we have∣∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx− n−1∑ k=1 k [ 1 + (−1)k ] (b−a)k 2k+1 (k + 1)! f(k) ( a + b 2 )∣∣∣∣∣∣ ≤ (b−a)n 2n+s/q+1n! (∫ 1 0 (1 − t)n−1 (n− 1 + t) dt )1−1/q × {(∫ 1 0 (1 − t)n−1 (n− 1 + t) [ (1 − t)s ∣∣∣f(n) (a)∣∣∣q + (1 + t)s ∣∣∣f(n) (b)∣∣∣q]dt)1/q + (∫ 1 0 (1 − t)n−1 (n− 1 + t) [ (1 − t)s ∣∣∣f(n) (b)∣∣∣q + (1 + t)s ∣∣∣f(n) (a)∣∣∣q]dt)1/q } . (2.3) Since ∫ 1 0 (1 − t)n−1 (n− 1 + t) dt = n n + 1 , ∫ 1 0 (1 − t)n−1 (n− 1 + t) (1 − t)s dt = n2 + s (n− 1) (n + s) (n + s + 1) , by using the property B (x,y + 1) = y x + y B (x,y) of the Euler Beta function and Lemma 2.2, we have∫ 1 0 (1 − t)n−1 (n− 1 + t) (1 + t)s dt ≤ ∫ 1 0 (1 − t)n−1 (n− 1 + t) (1 + ts) dt = n n + 1 + nB (s + 1,n) −B (s + 1,n + 1) = n n + 1 + nB (s + 1,n) − n n + s + 1 B (s + 1,n) = n n + 1 + n (n + s) n + s + 1 B (s + 1,n) . From the above facts and the inequality (2.3), we get the required inequality (2.2). This completes the proof of the Theorem. � The following corollaries are direct consequences of Theorem 2.1. Corollary 2.1. Under the assumptions of Theorem 2.1, if q = 1, we have the inequality∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx − n−1∑ k=1 k [ 1 + (−1)k ] (b−a)k 2k+1 (k + 1)! f(k) ( a + b 2 )∣∣∣∣∣∣ ≤ (b−a) n 2n+s+1n! × ( n2 + s (n− 1) (n + s) (n + s + 1) + n n + 1 + n (n + s) n + s + 1 B (s + 1,n) )[∣∣∣f(n) (a)∣∣∣ + ∣∣∣f(n) (b)∣∣∣] , (2.4) where B (α,β) is the Euler Beta function defined as in Theorem 2.1. INEQUALITIES FOR n-TIMES DIFFERENTIABLE s-CONVEX FUNCTIONS 123 Corollary 2.2. Under the assumptions of Theorem 2.1, if n = 1, we have the inequality∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx ∣∣∣∣∣ ≤ (b−a) ( 1 2 )3+s/q−1/q     ∣∣∣f′ (a)∣∣∣q (1 + s) (2 + s) + ( 1 2 + 1 2 + s )∣∣∣f′ (b)∣∣∣q   1/q +  (1 2 + 1 2 + s )∣∣∣f′ (a)∣∣∣q + ∣∣∣f′ (b)∣∣∣q (1 + s) (2 + s)   1/q   . (2.5) Corollary 2.3. If we take q = 1 in Corollary 2.2, we have∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx ∣∣∣∣∣ ≤ (b−a) (s + 3) s + 1 ( 1 2 )3+s [∣∣∣f′ (a)∣∣∣ + ∣∣∣f′ (b)∣∣∣] . (2.6) Corollary 2.4. Suppose the assumptions of Theorem 2.1 are fulfilled and if n = 2, we have∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx ∣∣∣∣∣ ≤ (b−a) 2 24+s/q ( 2 3 )1−1/q × {[ (s + 4) (s + 2) (s + 3) ∣∣∣f′′ (a)∣∣∣q + (2 3 + 2 (s + 1) (s + 3) )∣∣∣f′′ (b)∣∣∣q]1/q + [ (s + 4) (s + 2) (s + 3) ∣∣∣f′′ (b)∣∣∣q + (2 3 + 2 (s + 1) (s + 3) )∣∣∣f′′ (a)∣∣∣q]1/q } . (2.7) Corollary 2.5. If q = 1 in Corollary 2.4, we have∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx ∣∣∣∣∣ ≤ (b−a)2 24+s ( 2 3 + s2 + 7s + 8 (s + 1) (s + 2) (s + 3) )[∣∣∣f′′ (a)∣∣∣ + ∣∣∣f′′ (b)∣∣∣] . (2.8) Theorem 2.2. Let f : I ⊂ R → R be an n-times differentiable function on I◦, n ∈ N. If f(n) ∈ L ([a,b]), where a, b ∈ I◦ with a < b. If ∣∣f(n)∣∣q is convex on [a,b] for q ∈ (1,∞), we have the inequality∣∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx− n−1∑ k=1 k [ 1 + (−1)k ] (b−a)k 2k+1 (k + 1)! f(k) ( a + b 2 )∣∣∣∣∣∣ ≤ (b−a)n [ n(2q−1)/(q−1) − (n− 1)(2q−1)/(q−1) ]1−1/q 2n+s/q+1n! ( q − 1 2q − 1 )1−1/q ×   [ ∣∣f(n) (a)∣∣q nq −q + s + 1 + ( 1 nq −q + 1 + B (s + 1,nq −q + 1) )∣∣∣f(n) (b)∣∣∣q ]1/q + [ ∣∣f(n) (b)∣∣q nq −q + s + 1 + ( 1 nq −q + 1 + B (s + 1,nq −q + 1) )∣∣∣f(n) (a)∣∣∣q ]1/q  . (2.9) where B (α,β) is the Euler Beta function defined as in Theorem 2.1. 124 LATIF, DRAGOMIR AND MAMONAIT Proof. Using Lemma 2.1, by using the first inequality in Lemma 2.2, the Hölder inequality and con- vexity of ∣∣f(n)∣∣q on [a,b], we have ∣∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx− n−1∑ k=1 k [ 1 + (−1)k ] (b−a)k 2k+1 (k + 1)! f(k) ( a + b 2 )∣∣∣∣∣∣ ≤ (b−a)n 2n+s/q+1n! (∫ 1 0 (n− 1 + t)q/(q−1) dt )1−1/q × {(∫ 1 0 (1 − t)q(n−1) [ (1 − t)s ∣∣∣f(n) (a)∣∣∣q + (1 + ts) ∣∣∣f(n) (b)∣∣∣q]dt)1/q + (∫ 1 0 (1 − t)q(n−1) [ (1 − t)s ∣∣∣f(n) (b)∣∣∣q + (1 + ts) ∣∣∣f(n) (a)∣∣∣q]dt)1/q } . (2.10) By simple computations, we observe that∫ 1 0 (n− 1 + t)q/(q−1) dt = ( q − 1 2q − 1 )[ n(2q−1)/(q−1) − (n− 1)(2q−1)/(q−1) ] , ∫ 1 0 (1 − t)q(n−1)+s dt = 1 nq −q + s + 1 and ∫ 1 0 (1 − t)q(n−1) (1 + ts) dt = 1 nq −q + 1 + B (s + 1,nq −q + 1) . Using the above results in (2.10), we obtain the required result. This completes the proof of the theorem. � Corollary 2.6. Suppose the assumptions of Theorem 2.2 are satisfied and if n = 1, we have the inequality∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx ∣∣∣∣∣ ≤ (b−a)22+s/q ( q − 1 2q − 1 )1−1/q ( 1 s + 1 )1/q × {[∣∣∣f′ (a)∣∣∣q + (s + 2) ∣∣∣f′ (b)∣∣∣q]1/q + [∣∣∣f′ (b)∣∣∣q + (s + 2) ∣∣∣f′ (a)∣∣∣q]1/q} . (2.11) Corollary 2.7. Under the assumptions of Theorem 2.2, if n = 2, we have the inequality∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx ∣∣∣∣∣ ≤ (b−a)2 [ 2(2q−1)/(q−1) − 1 ]1−1/q 24+s/q ( q − 1 2q − 1 )1−1/q × {[ 1 q + s + 1 ∣∣∣f′′ (a)∣∣∣q + ( 1 q + 1 + B (s + 1,q + 1) )∣∣∣f′′ (b)∣∣∣q]1/q + [ 1 q + s + 1 ∣∣∣f′′ (b)∣∣∣q + ( 1 q + 1 + B (s + 1,q + 1) )∣∣∣f′′ (a)∣∣∣q]1/q } , (2.12) where B (α,β) is the Euler Beta function defined as in Theorem 2.1. INEQUALITIES FOR n-TIMES DIFFERENTIABLE s-CONVEX FUNCTIONS 125 Theorem 2.3. Let f : I ⊂ R → R be an n-times differentiable function on I◦, n ∈ N. If f(n) ∈ L ([a,b]), where a, b ∈ I◦ with a < b. If ∣∣f(n)∣∣q is convex on [a,b] for q ∈ (1,∞), we have the inequality∣∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx− n−1∑ k=1 k [ 1 + (−1)k ] (b−a)k 2k+1 (k + 1)! f(k) ( a + b 2 )∣∣∣∣∣∣ ≤ (b−a)n 2n+ s q +1n! ( q − 1 nq − 1 )1−1/q {[ P ∣∣∣f(n) (a)∣∣∣q + Q∣∣∣f(n) (b)∣∣∣q]1/q + [ P ∣∣∣f(n) (b)∣∣∣q + Q ∣∣∣f(n) (a)∣∣∣q]1/q} , (2.13) where P = nq+s+1B ( 1 n ; s + 1,q + 1 ) , Q = nq ( s + 2 s + 1 ) − 1 q + 1 −B (s + 1,q + 1) , B (α,β) is the Euler Beta function defined as in Theorem 2.1 and B (x; α,β) = ∫ x 0 tα−1 (1 − t)β−1 dt,α,β > 0, 0 ≤ x ≤ 1 is the incomplete Beta function. Proof. Using Lemma 2.1, the first inequality in Lemma 2.2, the Hölder inequality and convexity of∣∣f(n)∣∣q on [a,b], we have∣∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx− n−1∑ k=1 k [ 1 + (−1)k ] (b−a)k 2k+1 (k + 1)! f(k) ( a + b 2 )∣∣∣∣∣∣ ≤ (b−a)n 2n+s/q+1n! (∫ 1 0 (1 − t)q(n−1)/(q−1) dt )1−1/q × {(∫ 1 0 (n− 1 + t)q [ (1 − t)s ∣∣∣f(n) (a)∣∣∣q + (1 + ts) ∣∣∣f(n) (b)∣∣∣q]dt)1/q + (∫ 1 0 (n− 1 + t)q [ (1 − t)s ∣∣∣f(n) (b)∣∣∣q + (1 + ts) ∣∣∣f(n) (a)∣∣∣q]dt)1/q } . (2.14) By using the second inequality of Lemma 2.2 and simple computation, it is easy to observe that∫ 1 0 (1 − t)q(n−1)/(q−1) dt = q − 1 nq − 1 , ∫ 1 0 (n− 1 + t)q (1 − t)s dt = nq+s+1 ∫ 1 n 0 ts (1 − t)q dt = nq+s+1B ( 1 n ; s + 1,q + 1 ) = P and ∫ 1 0 (n− 1 + t)q (1 + ts) dt ≤ ∫ 1 0 (nq − (1 − t)q) (1 + ts) dt = nq ( s + 2 s + 1 ) − 1 q + 1 −B (s + 1,q + 1) = Q. Hence (2.13) follows from (2.14) and using the above results. This completes the proof of the theorem. � 126 LATIF, DRAGOMIR AND MAMONAIT Corollary 2.8. Suppose the assumptions of Theorem 2.3 are satisfied and if n = 1, we have the inequality∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx ∣∣∣∣∣ ≤ (b−a)22+s/q × {[ B (s + 1,q + 1) ∣∣∣f′ (a)∣∣∣q + (s + 2 s + 1 − 1 q + 1 −B (s + 1,q + 1) )∣∣∣f′ (b)∣∣∣q]1/q + [ B (s + 1,q + 1) ∣∣∣f′ (b)∣∣∣q + (s + 2 s + 1 − 1 q + 1 −B (s + 1,q + 1) )∣∣∣f′ (a)∣∣∣q]1/q } , (2.15) where B (α,β) is the Euler Beta function defined as in Theorem 2.1. Corollary 2.9. Suppose the assumptions of Theorem 2.3 are satisfied and if n = 2, we have the inequality∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx ∣∣∣∣∣ ≤ (b−a)2 24+ s q ( q − 1 2q − 1 )1−1/q {[ 2q+s+1B ( 1 2 ; s + 1,q + 1 )∣∣∣f′′ (a)∣∣∣q + ( 2q ( s + 2 s + 1 ) − 1 q + 1 −B (s + 1,q + 1) )∣∣∣f′′ (b)∣∣∣q]1/q + [ 2q+s+1B ( 1 2 ; s + 1,q + 1 )∣∣∣f′′ (b)∣∣∣q + ( 2q ( s + 2 s + 1 ) − 1 q + 1 − 2q+s+1B (s + 1,q + 1) )∣∣∣f′′ (a)∣∣∣q]1/q } , (2.16) where B (α,β) is the Euler Beta function defined as in Theorem 2.1 and B (x; α,β) is the incomplete Beta function defined as in Theorem 2.3. A different approach results in the following theorem. Theorem 2.4. Let f : I ⊂ R → R be a function such that f(n) exists on I◦ and f(n) ∈ L ([a,b]) for n ∈ N, where a, b ∈ I◦ with a < b. If ∣∣f(n)∣∣q is convex on [a,b] for q ∈ (1,∞), we have the inequality ∣∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx− n−1∑ k=1 k [ 1 + (−1)k ] (b−a)k 2k+1 (k + 1)! f(k) ( a + b 2 )∣∣∣∣∣∣ ≤ nn+1−1/q (b−a)n 2n+s/q+1n! ( 1 s + 1 )1/q [ B ( 1 n ; nq − 1 q − 1 , 2q − 1 q − 1 )]1−1/q × {[∣∣∣f(n) (a)∣∣∣q + (2s+1 − 1)∣∣∣f(n) (b)∣∣∣q]1/q + [( 2s+1 − 1 )∣∣∣f(n) (a)∣∣∣q + ∣∣∣f(n) (b)∣∣∣q]1/q} , (2.17) where B (x; α,β) = ∫ x 0 tα−1 (1 − t)1−β dt, 0 ≤ x ≤ 1,α > 0,β > 0 is the incomplete beta function. INEQUALITIES FOR n-TIMES DIFFERENTIABLE s-CONVEX FUNCTIONS 127 Proof. Using Lemma 2.1, the Hölder inequality and the convexity of ∣∣f(n)∣∣q on [a,b], we have∣∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx− n−1∑ k=1 k [ 1 + (−1)k ] (b−a)k 2k+1 (k + 1)! f(k) ( a + b 2 )∣∣∣∣∣∣ ≤ (b−a)n 2n+s/q+1n! (∫ 1 0 (1 − t)q(n−1)/(q−1) (n− 1 + t)q/(q−1) dt )1−1/q × {(∫ 1 0 [ (1 − t)s ∣∣∣f(n) (a)∣∣∣q + (1 + t)s ∣∣∣f(n) (b)∣∣∣q]dt)1/q + (∫ 1 0 [ (1 − t)s ∣∣∣f(n) (b)∣∣∣q + (1 − t)s ∣∣∣f(n) (a)∣∣∣q]dt)1/q } . (2.18) By simple computations of integrals, the inequality (2.17) follows from the inequality (2.18) and using the fact that∫ 1 0 (1 − t)q(n−1)/(q−1) (n− 1 + t)q/(q−1) dt = n nq+q−1 q−1 B ( 1 n ; nq − 1 q − 1 , 2q − 1 q − 1 ) . This completes the proof of the theorem. � Corollary 2.10. Suppose the assumptions of Theorem 2.4 are satisfied and if n = 1, we have∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx ∣∣∣∣∣ ≤ (b−a)22+s/q ( 1 s + 1 )1/q ( q − 1 2q − 1 )1−1/q × {[∣∣∣f′ (a)∣∣∣q + (2s+1 − 1)∣∣∣f′ (b)∣∣∣q]1/q + [(2s+1 − 1)∣∣∣f′ (a)∣∣∣q + ∣∣∣f′ (b)∣∣∣q]1/q} . (2.19) Proof. Proof follows from the fact that B ( 1; 1, 2q − 1 q − 1 ) = B ( 1, 2q − 1 q − 1 ) = ∫ 1 0 (1 − t) 2q−1 q−1 −1 dt = q − 1 2q − 1 . � Corollary 2.11. Under the assumptions of Theorem 2.4 and n = 2, we have the following inequality∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx ∣∣∣∣∣ ≤ (b−a)2 21+s/q+1/q ( 1 s + 1 )1/q [ B ( 1 2 ; 2q − 1 q − 1 , 2q − 1 q − 1 )]1−1/q × {[∣∣∣f′′ (a)∣∣∣q + (2s+1 − 1)∣∣∣f′′ (b)∣∣∣q]1/q + [(2s+1 − 1)∣∣∣f′′ (a)∣∣∣q + ∣∣∣f′′ (b)∣∣∣q]1/q} , (2.20) where B (x; α,β) is defined as in Theorem 2.4. 3. Applications to the Trapezoidal Formula Let d be a division of the interval [a,b], i.e. a = x0 < x1 < ... < xn−1 < xn = b, and consider the quadrature formula ∫ b a f(x)dx = T(f,d) + E(f,d), where T(f,d) = n−1∑ i=0 (xi+1 −xi) f (xi) + f (xi+1) 2 128 LATIF, DRAGOMIR AND MAMONAIT is the trapezoidal versions and E(f,d) is the associated error. Here, we derive some error estimates for the trapezoidal formula in terms of absolute values of the second derivative of f which may be better than those already exist in the literature. Theorem 3.1. Let f : I ⊆ R → R be a differentiable function on I◦ such that f ′′ ∈ L ([a,b]), where a,b ∈ I◦ with a < b. If ∣∣∣f′′∣∣∣q is s-convex on [a,b] for q ≥ 1, then for every division d of [a,b], we have |E(f,d)| ≤ 1 24+s/q ( 2 3 )1−1/q n−1∑ i=0 (xi+1 −xi) 3 × {[ (s + 4) (s + 2) (s + 3) ∣∣∣f′′ (xi)∣∣∣q + (2 3 + 2 (s + 1) (s + 3) )∣∣∣f′′ (xi+1)∣∣∣q]1/q + [ (s + 4) (s + 2) (s + 3) ∣∣∣f′′ (xi+1)∣∣∣q + (2 3 + 2 (s + 1) (s + 3) )∣∣∣f′′ (xi)∣∣∣q]1/q } . (3.1) Proof. By applying Corollary 2.4 on the subinterval [xi,xi+1] (i = 0, 1, . . . ,n− 1) of the division d, we have |E(f,d)| = ∣∣∣∣∣ n−1∑ i=0 { (xi+1 −xi) f (xi) + f (xi+1) 2 − ∫ xi+1 xi f (x) dx }∣∣∣∣∣ ≤ n−1∑ i=0 (xi+1 −xi) ∣∣∣∣f (xi) + f (xi+1)2 − 1xi+1 −xi ∫ xi+1 xi f (x) dx ∣∣∣∣ ≤ 1 24+s/q ( 2 3 )1−1/q n−1∑ i=0 (xi+1 −xi) 3 × {[ (s + 4) (s + 2) (s + 3) ∣∣∣f′′ (xi)∣∣∣q + (2 3 + 2 (s + 1) (s + 3) )∣∣∣f′′ (xi+1)∣∣∣q]1/q + [ (s + 4) (s + 2) (s + 3) ∣∣∣f′′ (xi+1)∣∣∣q + (2 3 + 2 (s + 1) (s + 3) )∣∣∣f′′ (xi)∣∣∣q]1/q } . (3.2) � Corollary 3.1. Under the assumptions of Theorem 3.1, for q = 1, we have |E(f,d)| ≤ 1 24+s ( 2 3 + s2 + 7s + 8 (s + 1) (s + 2) (s + 3) ) × n−1∑ i=0 (xi+1 −xi) 3 [∣∣∣f′′ (xi)∣∣∣ + ∣∣∣f′′ (xi+1)∣∣∣] . (3.3) Theorem 3.2. Let f : I ⊆ R → R be a differentiable function on I◦ such that f ′′ ∈ L ([a,b]), where a,b ∈ I◦ with a < b. If ∣∣∣f′′∣∣∣q is s-convex on [a,b] for q > 1, then for every division d of [a,b], we have |E(f,d)| ≤ [ 2(2q−1)/(q−1) − 1 ]1−1/q 24+s/q ( q − 1 2q − 1 )1−1/q n−1∑ i=0 (xi+1 −xi) 3 × {[ 1 q + s + 1 ∣∣∣f′′ (xi)∣∣∣q + ( 1 q + 1 + B (s + 1,q + 1) )∣∣∣f′′ (xi+1)∣∣∣q]1/q + [ 1 q + s + 1 ∣∣∣f′′ (xi+1)∣∣∣q + ( 1 q + 1 + B (s + 1,q + 1) )∣∣∣f′′ (xi)∣∣∣q]1/q } , (3.4) where B (α,β) is the Euler Beta function defined as in Theorem 2.1. INEQUALITIES FOR n-TIMES DIFFERENTIABLE s-CONVEX FUNCTIONS 129 Proof. It follows from Corollary 2.7. � Theorem 3.3. Let f : I ⊆ R → R be a differentiable function on I◦ such that f ′′ ∈ L ([a,b]), where a,b ∈ I◦ with a < b. If ∣∣∣f′′∣∣∣q is s-convex on [a,b] for q > 1, then for every division d of [a,b], we have |E(f,d)| ≤ 1 24+ 1 q ( q − 1 2q − 1 )1−1/q n−1∑ i=0 (xi+1 −xi) 3 × {[ 2q+s+1B ( 1 2 ; s + 1,q + 1 )∣∣∣f′′ (xi)∣∣∣q + ( 2q ( s + 2 s + 1 ) − 1 q + 1 −B (s + 1,q + 1) )∣∣∣f′′ (xi+1)∣∣∣q]1/q + [ 2q+s+1B ( 1 2 ; s + 1,q + 1 )∣∣∣f′′ (xi+1)∣∣∣q + ( 2q ( s + 2 s + 1 ) − 1 q + 1 − 2q+s+1B (s + 1,q + 1) )∣∣∣f′′ (xi)∣∣∣q]1/q } , (3.5) where B (α,β) is the Euler Beta functions and B (x; α,β) is the incomplete beta function defined as in Theorem 2.3. Proof. It is a direct consequence of Corollary 2.9. � Theorem 3.4. Let f : I ⊆ R → R be a differentiable function on I◦ such that f ′′ ∈ L ([a,b]), where a,b ∈ I◦ with a < b. If ∣∣∣f′′∣∣∣q is s-convex on [a,b] for q > 1, then for every division d of [a,b], we have |E(f,d)| ≤ 1 21+s/q+1/q ( 1 s + 1 )1/q [ B ( 1 2 ; 2q − 1 q − 1 , 2q − 1 q − 1 )]1−1/q × n−1∑ i=0 (xi+1 −xi) 3 {[∣∣∣f′′ (xi)∣∣∣q + (2s+1 − 1)∣∣∣f′′ (xi+1)∣∣∣q]1/q + [( 2s+1 − 1 )∣∣∣f′′ (xi)∣∣∣q + ∣∣∣f′′ (xi+1)∣∣∣q]1/q} , (3.6) where B (x; α,β) is the incomplete beta function defined as in Theorem 2.3. Proof. The proof follows by using Corollary 2.9. � 4. Applications to the Special Means Now, we consider applications of our results to special means. We consider the means for positive real numbers a, b ∈ R+. We consider (1) The arithmetic mean: A (a,b) = a + b 2 ; a,b > 0. (2) Generalized log-mean: Lp (a,b) = [ bp+1 −ap+1 (p + 1) (b−a) ]1 p ; a, b > 0, p ∈ R\{−1, 0} , a 6= b. Now we apply our results from Section 2 to give some inequalities for special means. It is shown in [4] that the function f : [0,∞) → R defined by f (t) =   a, t = 0, bts + c, t > 0, where s ∈ (0, 1), a, b, c ∈ R with 0 ≤ c ≤ a, b ≥ 0, is s-convex on [0,∞). Hence, for a = c = 0, b ≥ 0, the function f (t) = bts is an s-convex function on [0,∞), where s ∈ (0, 1). 130 LATIF, DRAGOMIR AND MAMONAIT Theorem 4.1. For a, b ∈ R+, a < b, 0 < s < 1 and q ∈ [1,∞), we have∣∣∣A(as+2,bs+2)−Ls+2s+2 (as+2,bs+2)∣∣∣ ≤ (b−a)2 23+s ( 2 3 (s + 2) (s + 1) + s2 + 7s + 8 s + 3 ) A (as,bs) . (4.1) Proof. Let f (x) = xs+2, x ∈ R+. Then ∣∣∣f′′ (x)∣∣∣ = (s + 2) (s + 1) xs is an s-convex function on R+. Applying Corollary 2.5, we obtain the required result. � Theorem 4.2. For a, b ∈ R+, a < b, 0 < s < 1 and q ∈ (1,∞), we have∣∣∣A(as/q+1,bs/q+1)−Ls/q+1s/q+1 (as/q+1,bs/q+1)∣∣∣ ≤ (b−a) 22+s/q ( q − 1 2q − 1 )1−1/q ( 1 s + 1 )1/q (s/q + 1) × { [2A (as,bs) + (s + 1) bs] 1/q + [2A (as,bs) + (s + 1) as] 1/q } . (4.2) Proof. Let f (x) = xs/q+1, x ∈ R+. Then ∣∣∣f′ (x)∣∣∣q = [(s/q + 1)]q xs is an s-convex function on R+. Applying Corollary 2.6, we obtain the required result. � Theorem 4.3. For a, b ∈ R+, a < b, 0 < s < 1 and n, q ∈ N, n,q > 1, we have∣∣∣A(as/q+1,bs/q+1)−Ls/q+1s/q+1 (as/q+1,bs/q+1)∣∣∣ ≤ (b−a) 22+s/q−1/q ( 1 s + 1 )1/q ( q − 1 2q − 1 )1−1/q (s/q + 1) × { [A (as,bs) + (2s − 1) bs]1/q + [(2s − 1) as + A (as,bs)]1/q } . (4.3) Proof. Let f (x) = xs/q+1, x ∈ R+. Then ∣∣∣f′ (x)∣∣∣q = [(s/q + 1)]q xs is an s-convex function on R+. Applying Corollary 2.10, we obtain the required result. � Remark 4.1. Many other interesting inequalities for means can be obtained by applying the other results to some suitable s-convex functions, however the details are left to the interested reader. Remark 4.2. For s = 1, we get some of the results from [12]. References [1] M. W. Alomari, M. Darus and U. S. Kirmaci, Some inequalities of Hadamard type inequalities for s-convex , Acts Math. Sci. Ser. B Engl. Ed. 31 (2011), no. 4, 1643-1652. [2] S. 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Comp., 217 (2011) 5171–5176. [14] C. E. M. Pearce and J. Pečarić, Inequalities for differentiable mappings with application to special means and quadrature formulae, Appl. Math. Lett., 13(2) (2000), 51–55. [15] Shan Peng, Wei Wei and JinRong Wang, On the Hermite-Hadamard Inequalities for Convex Functions via Hadamard Fractional Integrals Facta Universitatis (NIŠ) Ser. Math. Inform. 29 (1) (2014), 55-75. 1School of Computer Science and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa 2Mathematics, College of Engineering and Science, Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia; School of Computer Science and Applied Mathematics, University of the Witwater- srand, Private Bag 3, Wits 2050, Johannesburg, South Africa ∗Corresponding author: m amer latif@hotmail.com 1. Introduction 2. Main Results 3. Applications to the Trapezoidal Formula 4. Applications to the Special Means References