On strongly starlike and convex functions of order alpha and

Abstract

In this note we investigate the inclusion relationship between the class of strongly starlike functions of order alpha and type beta and the class of strongly convex functions of order alpha and type beta which are subclass of normalized analytic functions defined on the unit disk. Some applications of our main result are also presented which contains various classical results for the typical subclasses of starlike and convex functions.

Results & Lemmas (6)

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Theorem 1. Theorem 1. K(γ(α), β) ⊂S∗(α, β) for each α ∈(0, 1] and β ∈[0, 1). The above theorem includes Theorem A as the case when β = 0. We should…

Theorem 1. K(γ(α), β) ⊂S∗(α, β) for each α ∈(0, 1] and β ∈[0, 1). The above theorem includes Theorem A as the case when β = 0. We should notice the reader that this estimation is not sharp for each α ∈(0, 1] and β ∈[0, 1) (see also [5]). We will discuss about this problem in section 2 with the proof of Theorem 1. Our main theorem yields several applications which will be shown in the last section. 2. Proof of Theorem 1 Our proof relies on the following lemma which was obtained by the second auth

Lemma 2. Lemma 2. Tan−1α ≥ρ(α) for all α ∈(0, 1], where ρ is deﬁned by (1).

Lemma 2. Tan−1α ≥ρ(α) for all α ∈(0, 1], where ρ is deﬁned by (1).

Corollary 3. Corollary 3. K(α, β) ⊂S∗(α, β) for each α ∈(0, 1] and β ∈[0, 1).

Corollary 3. K(α, β) ⊂S∗(α, β) for each α ∈(0, 1] and β ∈[0, 1).

Theorem 1 Theorem 1 which is our desired inclusion. □

Theorem 1 which is our desired inclusion. □

Corollary 3 Corollary 3 yields the following property;

Corollary 3 yields the following property;

Corollary 4. Corollary 4. If z f ′(z) ∈S∗(α, β), then f ∈S∗(α, β).

Corollary 4. If z f ′(z) ∈S∗(α, β), then f ∈S∗(α, β).

Function classes studied:

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