On the Univalence Criterion of a General Integral Operator
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Research Article On the Univalence Criterion of a General Integral Operator 2 ¨ ¨ Daniel Breaz1 and H. Ozlem Guney 1 2
Department of Mathematics, “1 Decembrie 1918” University, Alba Iulia 510009, Romania Department of Mathematics, Faculty of Science and Arts, University of Dicle, Diyarbakir 21280, Turkey
Correspondence should be addressed to Daniel Breaz, [email protected] Received 12 March 2008; Accepted 16 May 2008 Recommended by Paolo Ricci In this paper we considered an general integral operator and three classes of univalent functions for which the second order derivative is equal to zero. By imposing supplimentary conditions for these functions we proved some univalent conditions for the considered general operator. Also some interesting particullar results are presented. ¨ Copyright q 2008 D. Breaz and H. Ozlem Guney. This is an open access article distributed under ¨ the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction Let unit disk and let A denote the class of functions f of the form fz z a2 z2 a3 z3 · · ·
z ∈ U,
1.1
which are analytic in the open disk U and satisfy the conditions f0 f 0 − 1 0. Consider S {f ∈ A : f are univalent functions in U}. Let A2 be the subclass of A consisting of functions of the form fz z
∞ ak zk .
1.2
k3
Let T be the univalent subclass of A which satisfies z2 f z − 1 < 1 z ∈ U. fz2
1.3
2
Journal of Inequalities and Applications
Let T2 be the subclass of T for which f 0 0. Let T2,μ be the subclass of T2 consisting of functions of the form 1.2 which satisfy z2 f z 1.4 − 1 ≤ μ z ∈ U fz 2 for some μ 0 < μ ≤ 1, and let us denote T2,1 ≡ T2 . Furthermore, for some real p with 0 < p ≤ 2 we define a subclass Sp of A consisting of all functions fz which satisfy z 1.5 fz ≤ p z ∈ U. In 1, Singh has shown that if fz ∈ Sp, then fz satisfies z2 f z − 1 ≤ p|z|2 z ∈ U. fz 2
1.6
Ahlfors 2 and Becker 3 had obtained the following univalence criterion. Theorem 1.1. Let c be a complex number, |c| ≤ 1, c / − 1. If fz z a2 z2 · · · is a regular function in U and 2 2 zf z 1.7 c|z| 1 − |z| ≤1 f z for all z ∈ U, then the function f is regular and univalent in U. In 4, Pescar had obtained the following theorem. Theorem 1.2 see 4. Let β be a complex number, Re β > 0, c a complex number, |c| ≤ 1, c / − 1, 2 and hz z a2 z · · · a regular function in U. If 2β 2β zh z 1.8 ≤1 c|z| 1 − |z| βh z for all z ∈ U, then the function z
1/β β−1 Fβ z β t h tdt z ···
1.9
0
is regular and univalent in U. Lemma 1.3 the general Schwarz lemma 5. Let the function fz be regular in the disk UR {z ∈ C; |z| < R}, with |fz| < M for fixed M. If fz has one zero with multiplicity order
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