Abstract
For every $q\in(0,1)$ and $0\le α<1$ we define a class of analytic functions, the so-called $q$-starlike functions of order $α$, on the open unit disk. We study this class of functions and explore some inclusion properties with the well-known class of starlike functions of order $α$. The paper is also devoted to the discussion on the Herglotz representation formula for analytic functions $zf'(z)/f(z)$ when $f(z)$ is $q$-starlike of order $α$. As an application we also discuss the Bieberbach conj
Results & Lemmas (10)
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Theorem 1.1.
Theorem 1.1. Let f ∈A. Then f ∈S∗ q (α) if and only if there exists a probability measure µ supported on the unit circle such that zf ′(z)…
Theorem 1.1. Let f ∈A. Then f ∈S∗ q (α) if and only if there exists a probability measure µ supported on the unit circle such that zf ′(z) f(z) = 1 + Z |σ|=1 σzF ′ q,α(σz)dµ(σ) where (3) Fq,α(z) = ∞ X
Theorem 1.3.
Theorem 1.3. Let (4) Gq,α(z):= z exp[Fq,α(z)] = z + ∞ X n=2 cnzn. Then Gq,α ∈S∗ q (α). Moreover, if f(z) = z + P∞ n=2 anzn ∈S∗ q (α), then…
Theorem 1.3. Let (4) Gq,α(z) := z exp[Fq,α(z)] = z + ∞ X n=2 cnzn. Then Gq,α ∈S∗ q (α). Moreover, if f(z) = z + P∞ n=2 anzn ∈S∗ q (α), then |an| ≤cn with equality holding for all n if and only if f is a rotation of Gq,α.
Proposition 2.1.
Proposition 2.1. Let f ∈S∗ q (α). Then there exists a unique function g ∈S∗ q such that (5) z(Dqf)(z) f(z) −α 1 −α = z(Dqg)(z) g(z) or…
Proposition 2.1. Let f ∈S∗ q (α). Then there exists a unique function g ∈S∗ q such that (5) z(Dqf)(z) f(z) −α 1 −α = z(Dqg)(z) g(z) or f(qz) −αqf(z) (1 −α)f(z) = g(qz) g(z) . holds. Similarly, for a given function g ∈S∗
Theorem 2.2.
Theorem 2.2. Let f ∈A. Then f ∈S∗ q (α) if and only if
Theorem 2.2. Let f ∈A. Then f ∈S∗ q (α) if and only if
Corollary 2.3.
Corollary 2.3. The class S∗ q (α) satisfies the inclusion relation q<p<1 S∗ p(α) ⊂S∗ q (α) and 0<q<1 S∗ q (α) = S∗(α).
Corollary 2.3. The class S∗ q (α) satisfies the inclusion relation \ q<p<1 S∗ p(α) ⊂S∗ q (α) and \ 0<q<1 S∗ q (α) = S∗(α).
Lemma 2.4.
Lemma 2.4. If h ∈Bq then the infinite product Q∞ n=0 ((1−α)h(zqn)+αq)/q converges uniformly on compact subsets of D.
Lemma 2.4. If h ∈Bq then the infinite product Q∞ n=0{((1−α)h(zqn)+αq)/q} converges uniformly on compact subsets of D.
Lemma 2.5.
Lemma 2.5. If h ∈B0 q then the infinite product Q∞ n=0 ((1 −α)h(zqn) + αq)/q con- verges uniformly on compact subsets of D to a nonzero…
Lemma 2.5. If h ∈B0 q then the infinite product Q∞ n=0{((1 −α)h(zqn) + αq)/q} con- verges uniformly on compact subsets of D to a nonzero function in H(D) with no zeros. Furthermore, the function (6) f(z) = z Q∞ n=0{((1 −α)h(zqn) + αq)/q}
Lemma 2.6.
Lemma 2.6. A function g ∈B0 q,α if and only if it has the representation (7) g(z) = exp ln q 1 −α(1 −q) p(z) , where p(z) belongs to…
Lemma 2.6. A function g ∈B0 q,α if and only if it has the representation (7) g(z) = exp ln q 1 −α(1 −q) p(z) , where p(z) belongs to the class P = {p : p ∈H(D), p(0) = 1 and Re {p(z)} ≥0 for z ∈D}.
Theorem 2.7.
Theorem 2.7. The mapping ρ: S∗ q (α) →B0 q defined by ρ(f)(z) = f(qz) f(z) −αq 1 −α is a bijection.
Theorem 2.7. The mapping ρ : S∗ q (α) →B0 q defined by ρ(f)(z) = f(qz) f(z) −αq 1 −α is a bijection.
Theorem 4.1.
Theorem 4.1. Let a, b, c be non-negative real numbers with 0 < 1 −aq < 1 −cq and 0 < 1 −b < 1 −c. For 0 < q < 1 and r ∈(0, 1], the function…
Theorem 4.1. Let a, b, c be non-negative real numbers with 0 < 1 −aq < 1 −cq and 0 < 1 −b < 1 −c. For 0 < q < 1 and r ∈(0, 1], the function z 7→zΦ[a, b; c; q, rz] has the order of q-starlikeness σq(zΦ[a, b; c; q, rz]) = 1 + ρq(1 −a)(1 −b) (1 −c)(1 −q) Φ[aq, bq; cq; q, ρ] Φ[a, b; c; q, ρ] where ρ = −r if q(1 −a) a(1 −q) =: s > 0 and ρ = r if s < 0. In particular, we have 1 + sρ 1 −ρ ≤σq(zΦ[a, b; c; q, rz]) ≤1 + ρs(1 −b) 2(1 −c) .
Function classes studied:
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