Ularu, Nicoleta; Breaz, Daniel Univalence criterion and convexity for an integral operator. (English) Zbl 1251.30029 Appl. Math. Lett. 25, No. 3, 658-661 (2012). Let \(f\) and \(g\) be normalized analytic functions defined on the open unit disk of the complex plane satisfying the conditions \[ |z^2f'(z)/f^2(z) -1|<1,\quad |f''(z)/f'(z)|<1\quad\text{and}\quad |g(z)|<M \] for some \(M>0\). For a complex number \(\alpha\) with non-negative real part satisfying \(2\sqrt{3}(1+2M^2)|\alpha|\leq 9\), the authors show that the integral \[ F(z):=\int_0^z[f'(t)\exp(g(t))]^\alpha dt \] is univalent in the unit disk. They also determine the order of convexity of the function \(F\) when \(f\) and \(g\) satisfy the conditions \[ |f'(z) (z/f(z))^\mu -1|<1-\beta,\quad |f''(z)/f'(z)|<1\quad\text{and}\quad |g(z)|<M. \] Reviewer: V. Ravichandran (Delhi) Cited in 1 ReviewCited in 4 Documents MSC: 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) Keywords:univalent function; convexity; univalence; integral operator PDFBibTeX XMLCite \textit{N. Ularu} and \textit{D. Breaz}, Appl. Math. Lett. 25, No. 3, 658--661 (2012; Zbl 1251.30029) Full Text: DOI References: [1] Frasin, B. A.; Jahangiri, J., A new and comprehensive class of analytic functions, Anal. Univ. Oradea, Fasc. Math., XV, 59-62 (2008) · Zbl 1199.30068 [2] Frasin, B. A.; Darus, M., On certain analytic univalent functions, Internat. J. Math. and Math. Sci., 25, 5, 305-310 (2001) · Zbl 0973.30013 [3] Becker, J., Löwnersche differentialgleichung und quasikonform fortsetzbare schlichte Funktionen, J. Reine Angew. Math., 255, 23-43 (1972) · Zbl 0239.30015 [4] Ozaki, S.; Nunokawa, M., The Schwartzian derivative and univalent functions, Proc. Amer. Math. Soc., 33, 2, 392-394 (1972) · Zbl 0233.30011 [5] Mayer, O., The Functions Theory of One Variable Complex (1981), Bucureşti This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.