Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Sufficient Univalence Conditions for Analytic Functions | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 86493 5 pages doi 2007 86493 Research Article Sufficient Univalence Conditions for Analytic Functions Daniel Breaz and Nicoleta Breaz Received 30 October 2007 Accepted 4 December 2007 Recommended by Narendra Kumar K. Govil We consider a general integral operator and the class of analytic functions. We extend some univalent conditions of Becker s type for analytic functions using a general integral transform. Copyright 2007 D. Breaz and N. Breaz. 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 z e C Izl 1 be the unit disk let denote the class of the functions f of the form f z z a2z2 a3z3 z e which are analytic in the open disk and let satisfy the condition f 0 f 0 - 1 0. Consider y f e f is univalent functions in . In 1 Pescar needs the following theorem. Theorem 1 . Let c and ft be complex numbers with Re ft 0 c 1 and c - 1 and let h z z a2z2 be a regular function in c z 2ft 1 -lzl2ft zh z I 1 c z u z fth z I 1 for all the z e u then the function Fft z ftJztft-1h t dt ft z is regular and univalent in u. 2 Journal of Inequalities and Applications In 2 Ozaki and Nunokawa give the next result. Theorem 2 . Let f e satisfy the following condition I z2f z I ff - Is1 L4 for all z e u then f is univalent in u. Lemma The Schwarz lemma 3 4 . Let the analytic function f be regular in the unit disk and let f 0 0 . If I f z 1 then f z z for all z e u where the equality can hold only if I f z I Kz and K 1. In 5 Seenivasagan and Breaz consider for f e 2 i 1 2 . n and a1 a2 . an f e C the integral operator F .n f z if P-IÍ pr 0 i 1 1 When a a for all i 1 2 . n Fa1 a2 . an f z becomes the integral operator Fa f z considered in 6 . 2. Main results Theorem . Let
