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Remarks on the certain subclass of univalent functions

Piejko K., Trojnar-Spelina L.

Journal of Mathematics and Applications

2008

Vol. 30

125-130

We investigate the family LP α (α ∈ (-π, π]) of functions [wzór] that are analytic in the unit disk with the property that the domain of values [wzór] is the parabolic region (Imw) ² < 2Rew - 1.We give inclusion theorems and bounds of Re �'(z) for this class.

Characterization of subclasses of univalent functions

Trojnar-Spelina L.

Demonstratio Mathematica

2005

Vol. 38, nr 1

35--41

We investigate the family of functions f(z) = z +(suma od n=2 do nieskończoności) anzn that are analytic in the unit disk with the property that the domain of values f(z) + 1+ei(alpha):2zf", (alpha is an element of (-pi,pi) is the parabolic region (Imw)2 < 2Rew - 1. Integral representation and convolution characterization are found and some coefficients bounds are given.

Differential subordinationfor domains bounded by hyperbolas

Kanas S., Lecko A.

Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka

1999

z. 23 [175]

61-70

Let the function pa,b,s-so with a, b real, nonnegative numbers and s, so is an element of R, be a conform al mapping of the unit disk onto a hyperbolic domain. We establish the conditions udner which the differential subordination of the type p(z) +[...]zp'(z) -< pa,b,s-so(^) yields p -< pa,b,s- so(z) yields p [...] pa,b,s in U. Some applications of obtained results are given.

Conic regions k-uniform convexity, II

Kanas S., Wiśniowska A.

Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka

1998

z. 22 [170]

65-78

In the previous paper, due to authors, the class of k-uniformly convex functions, (0 < k < oo), has been introduced. Mentioned denoted by kUCV, is a generalization of the class of uniformly convex functions introduced by Goodman, and studied by Ronning, Ma and Minda. This paper is a continuation of investigation of the class k-UCV for that the region of values of 1+ zf'(z)/f'(z), where/ f is an element of k-UCV, is a domain bounded by the conic curves, which kind depends of the parameter k. The estimates of If"(z)I, bounds of coefficients of k-uniformly convex functions, the sharp bounds on |anI, for n= 2,3,4, and the extremal functions which realize equality are given. Some particular examples of functions having the required properties are found.