Abstract
Several new inequalities for differentiable co-ordinated convex and concave functions in two variables which are related to the left side of Hermite- Hadamard type inequality for co-ordinated convex functions in two variables are obtained. Mathematics Subject Classification (2000): 26A51; 26D15
Highlights
The following definition is well known in literature: A function f: I → R, 0 = I ⊆ R, is said to be convex on I if the inequality f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y), holds for all x, y Î I and l Î [0, 1]
Motivated by notion given in [13], in the present article, we prove some new inequalities which give estimate between the middle and the leftmost terms in (1.2) for differentiable co-ordinated convex functions on rectangle from the plane R2
∂2 (1 − t)(1 − s) ∂s∂t f (ta + (1 − t)b, sc + (1 − s)d) dsdt 11
Summary
1. Introduction The following definition is well known in literature: A function f: I → R, 0 = I ⊆ R, is said to be convex on I if the inequality f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y), holds for all x, y Î I and l Î [0, 1]. A formal definition for co-ordinated convex functions may be stated as follows: Definition 1. [8]A function f: Δ ® R is said to be convex on the co-ordinates on Δ if the inequality f (tx + (1 − t)y, su + (1 − s)w) ≤ ts f (x, u) + t(1 − s)f (x, w) + s(1 − t)f (y, u) + (1 − t)(1 − s)f (y, w), holds for all t, s Î [0, 1] and (x, u), (y, w) Î Δ.
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