On Integral Inequalities of Hermite-Hadamard Type for s-Geometrically Convex Functions

and Applied Analysis 3 Theorem 1.6 5 , Theorem 4 . Letf : I → R0 be differentiable on I◦, a, b ∈ I, a < b, and f ′ ∈ L a, b . If |f ′ x | is s-convex on a, b for some s ∈ 0, 1 , and p, q ≥ 1, such that 1/q 1/p 1 then ∣ ∣ ∣ ∣ ∣ f ( a b 2 ) − 1 b − a ∫b a f x dx ∣ ∣ ∣ ∣ ∣ ≤ 2−1/p ∣f ′ a ∣ ∣q s 1 ∣ ∣f ′ a b /2 ∣ ∣q)1/q { s 1 s 2 } 2−1/p ∣f ′ b ∣ ∣q s 1 ∣ ∣f ′ a b /2 ∣ ∣q)1/q { s 1 s 2 } 2−1/p [( β s 1, 2 ∣ ∣f ′ a ∣ ∣q β s 2, 1 ∣ ∣ ∣ ∣f ′ ( a b 2 )∣ ∣ ∣ q)1/q ( β s 1, 2 ∣f ′ b ∣q β s 2, 1 ∣ ∣∣∣f ′ ( a b 2 )∣∣∣ q)1/q] . 1.7 Theorem 1.7 6 , Theorems 2.2–2.4 . Letf : I → R0 be differentiable on I◦, a, b ∈ I, a < b, and f ′ ∈ L a, b . i If |f ′ x | is s-convex on a, b for some s ∈ 0, 1 , then ∣∣∣∣ f ( a b 2 ) − 1 b − a ∫b a f x dx ∣∣∣∣ ≤ b − a 4 s 1 s 2 [∣∣f ′ a ∣ 2 s 1 ∣∣∣f ′ ( a b 2 )∣∣∣ ∣f ′ b ∣ ] ≤ ( 22−s 1 ) b − a 4 s 1 s 2 ∣f ′ a ∣ ∣f ′ b ∣. 1.8 ii If |f ′ x | p−1 p > 1 is a s-convex function on a, b for some s ∈ 0, 1 , then ∣∣∣∣ f ( a b 2 ) − 1 b − a ∫b a f x dx ∣∣∣∣ ≤ b − a 4 ( 1 p 1 )1/p( 1 s 1 )2/q [(( 21−s s 1 ∣f ′ a ∣∣q 21−s ∣f ′ b ∣∣q )1/q ( 21−s ∣f ′ a ∣q∣f ′ a ∣p/ p−1 ( 21−s s 1 ∣f ′ b ∣q )1/q] , 1.9