Further Inequalities for Power Series with Nonnegative Coefficients Via a Reverse of Jensen Inequality

Some inequalities for power series with nonnegative coefficients via a new reverse of Jensen inequality are given. Applications for some fundamental functions defined by power series are also provided.


Introduction
In 1994, Dragomir & Ionescu obtained the following reverse of Jensen's discrete inequality: Let Φ : I → R be a differentiable convex function on the interiorI of the interval I. If x i ∈I and w i ≥ 0 (i = 1, . . . , n) with W n := n i=1 w i = 1, then one has the inequality: In order to improve Grüss' discrete inequality, Cerone & Dragomir established in 2002 the following result [1]: provided ∞ < a ≤ a i ≤ A < ∞, and w i ≥ 0 (i = 1, . . . , n) with W n := n i=1 w i = 1.
In addition, if ∞ < b ≤ b i ≤ B < ∞, (i = 1, . . . , n) then we have the string of inequalities Utilising these results, we observe that if Φ is differentiable convex on a finite interval, say [m, M ] , then we have the inequalities: If the lateral derivatives Φ + (m) and Φ − (M ) are finite, then we also have Corollary 1.1. Let f (z) = ∞ n=0 a n z n be a power series with nonnegative coefficients and convergent on the open disk D (0, R) with R > 0 or R = ∞.
Motivated by the above results and utilizing a new reverse of Jensen inequality we provide in this paper other inequalities for power series with nonnegative coefficients. Applications for some fundamental functions are given as well.

Reverses of Jensen's Inequality
The following reverse of the Jensen's inequality holds: We also have the inequality By denoting we have for any t ∈ (m, M ) . Therefore we have the equality which by (15) and (17) produces the desired result (13). Since, obviously then by (15) and (17) we deduce the second inequality (14). The last part is clear.
If we write the inequalities (25) and (26) for these choices, we get the weighted inequalities Theorem 3.1. Let f (z) = ∞ n=0 a n z n be a power series with nonnegative coefficients and convergent on the open disk D (0, R) with R > 0 or R = ∞. If p > 1, 0 < α < R and 0 < x ≤ 1, then for t ∈ (0, 1) .
Proof. If 0 < α < R and m ≥ 1, then by (46) for x j = (x p ) j , we have This is equivalent to Since all series whose partial sums involved in the inequality (48) are convergent, then by letting m → ∞ in (48) we deduce (47).
Theorem 5.1. Let f (z) = ∞ n=0 a n z n be a power series with nonnegative coefficients and convergent on the open disk D (0, R) with R > 0 or R = ∞. If x ≤ 0, β > 0 with exp (βx) < R and 0 < α < R, then Proof. If 0 < α < R and m ≥ 1, then by (51) for x j = jx, we have Since all series whose partial sums involved in the inequality (53) are convergent, then by letting m → ∞ in (53) we deduce (52).