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Let Bp(α, β, λ; j) be the class consisting of functions f(z) = zp + Σ∞k=p+1 akzk, p ∈ N which satisfy Re ..., for some λ (λ < p!{α+(p-j)β+(p-j)(p-j-1)(β-α)/2}/(p-j)!) and j = 0, 1, ..., p, where p+1-j+2α/(β-α) > 0 or α = β = 1. The extreme points of Bp(α, β, λ; j) are determined and various sharp inequalities related to Bp(α, β, λ; j) are obtained. These include univalence criteria, coefficient bounds, growth and distortion estimates and bounds for certain linear operators. Furthermore, inclusion properties are investigated and estimates on λ are found so that functions of Bp(α, β, λ; j) are p-valent starlike in U. For instance, Re{zf′′ (z)} > (5 - 12 ln 2)/(44 - 48 ln 2) ≈ -0.309 is sufficient condition for any normalized analytic function f to be starlike in U. The results improve and include a number of known results as their special cases. [ABSTRACT FROM AUTHOR]

For normalized analytic functions f(z) with f(z)≠ 0 for 0 < |z| < 1, we introduce a univalence criterion defined by sharp inequality associated with the nth derivative of z/f(z), where n ∈ {3, 4, 5, . . .}. [ABSTRACT FROM AUTHOR]

Let A be the class of analytic functions in the open unit disk U. We define ϴα,β : A → A by (ϴα,βƒ)(z) ≔ Γ(2 - α)zαDαz)Γ(2 - β)zβDβ zƒ)z)), (α, β≠ 2, 3, 4 …), where Dγzƒ is the fractional derivative of ƒ of order γ. If α, β ϵ [ 0, 1], then a function ƒ in A is said to be in the class SPα,β if ϴα,βƒ is a parabolic starlike function. In this paper, several properties and characteristics of the class SPα,β are investigated. These include subordination, characterization and inclusions, growth theorems, distortion theorems, and class-preserving operators. Furthermore, sandwich theorem related to the fractional derivative is proved. [ABSTRACT FROM AUTHOR]