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Ma-Minda φ-classes studied in this paper:
Abstract
Several inclusions between the class of functions with positive real
part and the class of starlike univalent functions associated with lemniscate of
Bernoulli are obtained by making use of the well-known theory of differential
subordination. Further, these inclusions give sufficient conditions for normalized
analytic functions to belong to some subclasses of starlike functions. The results
also provide sharp version of some previously known results.
Mathematics Subject Classification (2010): 30C45.
Results & Lemmas (4)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Theorem 2.1.
Theorem 2.1. Let the function p be analytic in D, p(0) = 1 and 1+βzp′(z) ≺√1 + z. Then the following subordination results hold: (a) If β…
Theorem 2.1. Let the function p be analytic in D, p(0) = 1 and 1+βzp′(z) ≺√1 + z. Then the following subordination results hold: (a) If β ≥2( √ 2−1+log 2−log(1+ √ 2)) √ 2−1 ≈1.09116, then p(z) ≺√1 + z. (b) If β ≥2(1−log 2) 3−2 √ 2 ≈3.57694, then p(z) ≺ϕ0(z).
Lemma 2.2.
Lemma 2.2. [15, Theorem 3.4h, p. 132] Let q be analytic in D and let ψ and ν be analytic in a domain U containing q(D) with ψ(w) ̸= 0 when…
Lemma 2.2. [15, Theorem 3.4h, p. 132] Let q be analytic in D and let ψ and ν be analytic in a domain U containing q(D) with ψ(w) ̸= 0 when w ∈q(D).
Theorem 2.5.
Theorem 2.5. Let the function p be analytic in D, p(0) = 1 and 1 + βzp′(z)/p(z) ≺ √ 1 + z. Then the following subordination results hold:…
Theorem 2.5. Let the function p be analytic in D, p(0) = 1 and 1 + βzp′(z)/p(z) ≺ √ 1 + z. Then the following subordination results hold: (a) If β ≥2(log 2−1) log(2 √ 2−2) ≈3.26047, then p(z) ≺ϕ0(z). (b) If β ≥ 2( √ 2−1+log(2)−log( √ 2+1))
Theorem 2.8.
Theorem 2.8. Let the function p be analytic in D, p(0) = 1 and 1 + βzp′(z)/p2(z) ≺ √ 1 + z. Then the following subordination results hold…
Theorem 2.8. Let the function p be analytic in D, p(0) = 1 and 1 + βzp′(z)/p2(z) ≺ √ 1 + z. Then the following subordination results hold for sharp bound of β: (a) If β ≥4(1 + √ 2)(1 −log 2) ≈2.96323, then p(z) ≺ϕ0(z).
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