A study on the generalization of janowski functions in the unit disc

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Polatog̃lu Y., Bolcal M., Şen A., Yavuz E.

Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis, cilt.22, sa.1, ss.27-31, 2006 (Scopus) identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 22 Sayı: 1
  • Basım Tarihi: 2006
  • Dergi Adı: Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis
  • Derginin Tarandığı İndeksler: Scopus
  • Sayfa Sayıları: ss.27-31
  • Anahtar Kelimeler: Distortion theorem, Radius of starlikeness, Representation theorem, Starlike functions
  • İstanbul Kültür Üniversitesi Adresli: Evet

Özet

Let Ω be the class of functions w(z), |w(0) = 0, w(z)| < 1 regular in the unit disc D = {z : |z| < 1}. For arbitrarily fixed numbers A ∈ (-1, 1], B ∈ [-1, A), 0 ≤ α < 1 let P(,4, B, α) be the class of regular functions p(z) in D such that p(0) = 1, and which is p(z) ∈ P(A, B, α) if and only if p(z) = 1+[(1 - α) A+αB]w(z)/1+Bw(z) for some function w(z) ∈ Ω and every 5 ∈ D. In the present paper we apply the principle of subordination ([1], [3], [4], [5]) to give new proofs for some classical results concerning the class S*(A, B, α) of functions f(z) with f(0) = 0, f′(0) = 1, which are regular in D satisfying the condition: f(z) ∈ S*(A, B, α) if and only if z f′/f(z) = p(z) for some p(z) ∈ P(A, S, α) and for all z in D.