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On the order of starlikeness of hypergeometric functions. (English) Zbl 0598.30021

An analytic function f in a domain D is called starlike of order \(\gamma <1\) if and only if \[ f(0)=0,\quad f'(0)=1\quad and\quad Re[zf'(z)/f(z)]>\gamma,\quad z\in D. \] \(S^*_{\gamma}\) denotes the set of these functions. The authors estimate the order of starlikeness of the hypergeometric functions \(u(z)=z_ 2F_ 1(a,b;c: \rho z)\). Some interesting applications and a confluent case have also been given.
Reviewer: A.D.Wadhwa

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
33C05 Classical hypergeometric functions, \({}_2F_1\)
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[1] Abramowitz, M; Stegun, I, ()
[2] Kreyszig, E; Todd, J, The radius of univalence of the error function, Numer. math., 1, 78-89, (1959) · Zbl 0086.06203
[3] Lewis, J, Applications of a convolution theorem to Jacobi polynomials, SIAM J. math. anal., 10, 1110-1120, (1979) · Zbl 0498.33013
[4] Merkes, E; Scott, B.T, Starlike hypergeometric functions, (), 885-888 · Zbl 0102.06502
[5] Miller, S.S; Mocanu, P; Reade, M, The order of starlikeness of alpha-convex functions, Mathematica, 43, 25-30, (1978) · Zbl 0398.30008
[6] Ruscheweyh, St, Convolutions in geometric functions theory, Sém. math. sup., 83, (1982) · Zbl 0575.30008
[7] {\scSt. Ruscheweyh and D. Wilken}, Sharp estimates for certain Briot-Bouquet subordination, Rev. Roumaine de Math. Pures Appl., to appear. · Zbl 0585.30030
[8] Ruscheweyh, St; Schwittek, P, On real functions of bounded variation and an application to geometric function theory, Ann. univ. mariae Curie-skl., 36/37, 135-142, (1982/1983) · Zbl 0575.30008
[9] Sheil-Small, T; Silverman, H; Silvia, E, Convolution multipliers and starlike functions, J. anal. math., 41, 181-192, (1982) · Zbl 0526.30010
[10] Wall, H, Analytic theory of continued fractions, (1948), Van Nostrand Toronto · Zbl 0035.03601
[11] Wilken, D; Feng, J, A remark on convex and starlike functions, J. London math. soc., 21, 287-290, (1980) · Zbl 0431.30007
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