Milutin Obradović

On certain properties for some classes of starlike functions

Abstract Let f(z) = z + a2z2 + … be analytic in the unit disk U = {z:|z|⩽1}. By using the method of differential subordinations we give a criterion for a function f(z) to be in a certain class L ∗ [a,b] of starlike functions. For functions f(z)∈L ∗ [a,b] a subordination relation for (f(z) z) μ is also given.

Radius properties for subclasses of univalent functions

Summary A normalized analytic function f(z) = z + a2z2 + ··· (|z| < 1) is said to be in U (resp. P(2)) if for |z| < 1, It is known that P(2) ≠⊆ U ≠⊆ S, where S denotes the set of all normalized analytic functions that are univalent in |z| < 1. In this paper, we prove a general result which implies that We also show that if f ∊ S, then one has r−1f(rz) ∊ P(2) for 0 < r < r0, where r0 = 0.60629, correctly rounded to six decimal places, is the unique root of the equation 2r8 − 9r6 + 10r4 − 8r2 + 2 = 0.

On certain inequalities for some regular functions in |z|<1

In this paper we give some inequalities for regular functions f ( z ) = z + a 2 z 2 + … in | z | 1 especially for starlike and convex functions of order α , 0 ≤ α 1 , To some extent those inequalities are the generalisations and improvements of the previous results given by Bernardi [1]. Some interesting consequences are given, too.

Certain subclasses of Bazilevič functions of type α

Certain subclasses B(α,β) and B1(α,β) of Bazilevic functions of type α are introduced. The object of the present paper is to derive a lot of interesting properties of the classes B(α,β) and B1(α,β).

Product of univalent functions

Abstract Let S denote the class of functions f analytic and univalent in the unit disk | z | 1 normalized such that f ( 0 ) = 0 = f ′ ( 0 ) − 1 . In this article the authors discuss the radius of univalence of F ( z ) = g ( z ) h ( z ) / z when g and h belong to certain subsets of S . The paper concludes with the following conjecture. If g , h ∈ S , then F is univalent for | z | 1 / 3 and the number 1 / 3 cannot improved. The conjecture is shown to be true for some subclasses of S , e.g. the class of starlike functions, and the class U consisting of functions f ∈ A satisfying the functional inequality | f ′ ( z ) ( z f ( z ) ) 2 − 1 | 1 , ∣ z ∣ 1 . Some other related results are also presented.

SOME SUFFICIENT CONDITIONS FOR STRONGLY STARLIKENESS

We consider strongly starlikeness of order α of functions f( z)= z +an+1z n+1 +··· which are analytic in the unit disc and satisfy the condition of the form f � (z) z f( z) 1+µ − 1 <λ , 0 <µ< 1, 0 <λ< 1.

An application of differential subordinations and some criteria for univalency

By using the method of differential subordinations, we derive, among other results, some criteria for univalency in the unit disc.

TWO PARAMETER FAMILIES OF CLOSE-TO-CONVEX FUNCTIONS AND CONVOLUTION THEOREMS

Using a recently obtained coefficient condition for functions with positive real part by M. Obradović and S. Ponnusamy, we construct four different useful examples of two parameter families of close-to-convex functions. As a consequence of another well-known coefficient condition due to M. Obradović and S. Ponnusamy, we obtain a number of results concerning the convolution of two univalent functions. Using these results, we obtain criteria for combinations of hypergeometric functions to be univalent.

Necessary and sufficient conditions for univalent functions

Let 𝒜 be the class of analytic functions in the unit disc with the normalization f (0) = f ′(0) − 1 = 0. This article analyses various necessary and sufficient coefficient conditions for functions f ∈ 𝒜 of the form to be univalent. We present an interesting class of univalent functions associated with the zeta function and also pose an open problem.

Analytic functions of non-Bazilevič type and starlikeness

Two classes ℬ¯n(μ,α,λ) and 𝒫¯n(μ,α,λ) of analytic functions which are not Bazilevic type in the open unit disk 𝕌 are introduced. The object of the present paper is to consider the starlikeness of functions belonging to the classes ℬ¯n(μ,α,λ) and 𝒫¯n(μ,α,λ).