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  • 1 University of Gunma, Hoshikuki-cho 798-8, Chuou-Ward, Chiba, 260-0808, Japan
  • | 2 University of Rzeszów, ul. Prof. Pigonia 1, 35-310 Rzeszów, Poland
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Let L(f, r) denote the length of the closed curve which is the image of |z| = r < 1 under the mapping w = f(z). We establish some sufficient conditions for L(f, r) to be bounded and for f(z) to in the classes of strongly close-to-convex function of order α and to be strongly Bazilevič function of type β of order α. Moreover, we prove an inequality connected with the Fejér-Riesz's inequality.

  • [1]

    Brannan, D. A. and Kirwan, W. E., On some classes of bounded univalent Functions, J. London Math. Soc., 1(2) (1969), 431-443.

  • [2]

    Fejér, L. and Riesz, F., Uber einige funktionentheoretische Ungleichungen, Math. Z., 11 (1921), 305-314.

  • [3]

    Goodman, A. W., Univalent Functions, Vols. I and II, Mariner Publishing Co., Tampa, Florida (1983).

  • [4]

    Nunokawa, M. and Sokél, J., On some lenght problems for analytic functions, Osaka J. Math., 51 (2014), 695-707.

  • [5]

    Rogosinski, W., On the coefficients of subordinate functions, Proc. London Math. Soc., 48(2) (1943), 48-82.

  • [6]

    Stankiewicz, J., Quelques problèmes extrémaux dans les classes des fonctions a-angulairement étoilées, Ann. Univ. Mariae Curie-Sklodowska, Sect. A, 20 (1965), 59-75.

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    • Export Citation
  • [7]

    Stankiewicz, J., On a family of starlike functions, Ann. Univ. Mariae Curie-Sklodowska, Sect. A, 22-24 (1968/70), 175-181.