in this paper we obtain coefficient characterization‎, ‎extreme points and‎ ‎distortion bounds for the classes of harmonic $p-$valent functions‎ ‎defined by certain modified operator‎. ‎some of our results improve‎ ‎and generalize previously known results‎.

Making use of a certain fractional calculus operator, we introduce two new subclasses Mδ(p;λ, μ, η) and Tδ(p;λ, μ, η) of analytic and p−valent functions in the open unit disk. The results investigated exhibit the sufficiency conditions for a function to belong to each of these classes. Several geometric properties of such multivalent functions follow, and these consequences are also mentioned.

Let K = ( k j ) be an increasing sequence of integers, 0 , and E a class analytic in the disk D z : < 1 functions f such that for some p ∈ ℕ all derivatives are middle -valent . It is proved every function entire if only lim → ∞ ln - ∑ s +

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.

In this paper, sharp upper bounds for ∣∣ap+2 − ηa2p+1∣∣ and |ap+3| are derived for a class of Mocanu α-convex p-valent functions defined by an extended linear multiplier differential operator (LMDO) J δ p (λ, μ, l). Mathematics Subject Classification (2010): Primary 30C45, 30C50, Secondary 30C80.

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...

In this paper, we introduce the generalized integral operator Jp(σ, λ) and using this generalized integral operator, the new subclasses H n,m(b, σ, λ), Ln,m(b, σ, λ;μ), H n,m(b, σ, λ) and L n,m(b, σ, λ;μ) of the class of multivalent functions denoted by Tp(n) are defined. Further for functions belonging to these classes, certain properties of neighborhoods of functions of complex order are stud...