The purpose of this paper is to derive some subordination and superordination results for functions of the form f(z) = zp + ∞ ∑ k=p+1 akz k which are p−valent in the open unit disk U = {z ∈ C : |z| < 1} by using certain differential operator Aλ,p(α, β, μ)f(z) and integral operator F m p (ρ, θ)f(z). Some special cases are also considered.

Let J n+p−1 (α) (p ∈ N, n > −p, 0 α < p) denote the class of functions of the form f (z) = 1 z p + a 0 z p−1 + a 1 z p−2 + ··· which are regular and p-valent in the punctured unit disc U * = {z ∈ C : 0 < |z| < 1} and satisfy the condition Re (D n+p f (z)) (D n+p−1 f (z)) − (p + 1) < − p (n + p − 1) + α n + p , where D n+p−1 f (z) = 1 z p (1 − z) n+p * f (z). It is proved that J n+p (α) ⊂ J n+p−...

In the present investigation, the authors derive necessary and sufficient conditions for spirallikeness and convex spirallikeness of a suitably normalized meromorphic p-valent function in the punctured unit disk, using convolution. Also we give an application of our result to obtain a convolution condition for a class of meromorphic functions defined by a linear operator.

This review focuses on the development of ruthenium and flavin catalysts for environmentally benign oxidation reactions based on mimicking the functions of cytochrome P-450 and flavoenzymes, and low valent transition-metal catalysts that replace conventional acids and bases. Several new concepts and new types of catalytic reactions based on these concepts are described. (Communicated by Ryoji N...

Abstract. In this paper, we introduce and investigate the various properties and characteristics of the classMa,c(p;β;A,B) and its subclassM + a,c(p; β;A,B) of meromorphically p-valent functions of order β(pβ > 1), which are defined by means of the Liu-Srivastava operator. In particular, several inclusion relations, coefficients estimates, distortion theorems ,neighborhoods partial, Hadamard pr...