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Generalizations and refinements of Hermite-Hadamard’s inequality. (English) Zbl 1096.26014

The Hermite-Hadamard inequality can be easily extended to the case of twice differentiable functions f with bounded second derivative. Precisely, if γf '' Γ, then

3S 2 -2Γ 24(b-a) 2 1 b-a a b fdt-fa+b 23S 2 -2γ 24(b-a) 2


3S 2 -γ 12(b-a) 2 f(a)+f(b) 2-1 b-a a b fdtΓ 12(b-a) 2

The paper under review contains extensions of the Hermite-Hadamard inequality to the context of functions with bounded derivatives of nth order. For example, if f:[a,b] is an n-times differentiable function with γf (n) Γ, then it is proved that

(b-a) n+1 n!2 n S n +1+(-1) n 2(n+1)-1Γ(-1) n a b fdt
+ i=0 n-1 (b-a) n-i (n-i)!(-1) n+1 +(-1) i 2 n-i f (n-i-1) a+b 2(b-a) n+1 n!2 n S n +1+(-1) n 2(n+1)-1γ

where S n =f (n-1) (b)-f (n-1) (a) b-a· Further extensions are obtained via the concept of harmonic sequence of polynomials.

26D15Inequalities for sums, series and integrals of real functions
41A55Approximate quadratures