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Differential subordination and Bazilevič functions. (English) Zbl 0859.30024

Let p, λ, h and φ be analytic functions in the unit disc Δ. In this paper, the author proves that, under suitable assumptions on the above functions, the following relation holds: p(z)+λ(z)zp ' (z)h(z) implies p(z)φ(z)h(z) for zΔ, where denotes the subordination of functions.

Basing himself on this result he proves a sufficient condition for an analytic function to be starlike in Δ. These results are some extensions of the classical result of Sakaguchi and Libera.

MSC:
30C80Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable)
30C45Special classes of univalent and multivalent functions
References:
[1]Chichra P N, New subclasses of the class of close-to-convex functions,Proc. Am. Math. Soc. 62 (1977) 37–43 · doi:10.1090/S0002-9939-1977-0425097-1
[2]Hallenbeck D J and Ruscheweyh S, Subordination by convex functions,Proc. Am. Math. Soc. 52 (1975) 191–195 · doi:10.1090/S0002-9939-1975-0374403-3
[3]Krzyż J, A counter example concerning univalent function,Mat. Fiz. Chem. 2 (1962) 57–58
[4]Miller S S and Mocanu P T, Second order differential inequalities in the complex plane,J. Math. Anal. Appl. 65 (1978) 289–305 · Zbl 0379.34003 · doi:10.1016/0022-247X(78)90181-6
[5]Miller S S and Mocanu P T, Differential subordinations and Inequalities in the complex plane,J. Differ. Equ. 67 (1987) 199–211 · Zbl 0633.34005 · doi:10.1016/0022-0396(87)90146-X
[6]Miller S S and Mocanu P T, Marx-Strohhäcker differential subordinations systems,Proc. Am. Math. Soc. 99 (1987) 527–534
[7]Mocanu P T, Ripeanu D and Popovici M, Best bound for the argument of certain analytic functions with positive real part, Prepr., Babes-Bolyai Univ.,Fac. Math., Res. Semin. 5 (1986) 91–98
[8]Ponnusamy S and Karunakaran V, Differential Subordination and Conformal Mappings,Complex Variables: Theory and Appl. 11 (1989) 79–86
[9]Ponnusamy S, Differential Subordination and Starlike Functions,Complex variables: Theory and Appln. 19 (1992) 185–194
[10]Ponnusamy S, Convolution of Convexity under Univalent and Non-univalent Mappings, Internal Report (1990)
[11]Ruscheweyh S, Neighborhoods of univalent functions,Proc. Am. Math. Soc. 81 (1981) 521–527 · doi:10.1090/S0002-9939-1981-0601721-6
[12]Yoshikawa H and Yoshikai T, Some notes on Bazilevič functions,J. London Math. Soc. 20 (1979) 79–85 · Zbl 0409.30018 · doi:10.1112/jlms/s2-20.1.79