On an inequality of Diananda. Part II

On an Inequality of Diananda, Iii

We extend the results in part I, II on certain inequalities involving the generalized power means.

On an Inequality of Diananda - Part IV

On an inequality of Diananda. Part III

Let Mn,r(x) be the generalized weighted means Mn,r(x) = ( ∑n i=1 qix r i ) 1/r , where Mn,0(x) denotes the limit of Mn,r(x) as r → 0+, x = (x1,x2, . . . ,xn), and qi > 0 (1≤ i≤ n) are positive real numbers with ∑n i=1 qi = 1. In this paper, we let q =minqi and always assume n≥ 2, 0≤ x1 < x2 < ··· < xn. We define An(x)=Mn,1(x), Gn(x)=Mn,0(x),Hn(x)=Mn,−1(x) and we will writeMn,r forMn,r(x),An for...

An inequality related to $eta$-convex functions (II)

Using the notion of eta-convex functions as generalization of convex functions, we estimate the difference between the middle and right terms in Hermite-Hadamard-Fejer inequality for differentiable mappings. Also as an application we give an error estimate for midpoint formula.

On an Inequality of Diananda

where Pn,0(x) denotes the limit of Pn,r (x) as r → 0+ and where qi > 0, 1≤ i≤n, are positive real numbers with ∑n i=1qi = 1 and x = (x1,x2, . . . ,xn). In this note, we let q =minqi and always assume n≥ 2 and 0≤ x1 <x2 < ···<xn. We define An(x) = Pn,1(x), Gn(x) = Pn,0(x), and Hn(x) = Pn,−1(x) and we will write Pn,r for Pn,r (x), An for An(x), and similarly for other means when there is no risk ...