Volterra integral operators and logarithmic convexity.

On the Convexity of Certain Integral Operators

In this paper we consider the classes of starlike functions of order α, convex functions of order α and we study the convexity and α-order convexity for some general integral operators. Several corollaries of the main results are also considered.

Logarithmic Convexity of Area Integral Means for Analytic Functions Ii

For 0 < p < ∞ and −2 ≤ α ≤ 0 we show that the L integral mean on rD of an analytic function in the unit disk D with respect to the weighted area measure (1−|z|) dA(z) is a logarithmically convex function of r on (0, 1).

Logarithmic Convexity of Area Integral Means for Analytic Functions

We show that the L integral mean on rD of an analytic function in the unit disk D with respect to the weighted area measure (1 − |z|) dA(z), where −3 ≤ α ≤ 0, is a logarithmically convex function of r on (0, 1). We also show that the range [−3, 0] for α is best possible.

Spectral theory, Volterra integral operators and the Sturm-Liouville theorem

Convexity of integral operators of p-valent functions

In this paper, we consider two general p-valent integral operators for certain analytic functions in the unit disc U and give some properties for these integral operators on some classes of univalent functions.