Coefficient estimates for some classes of functions associated with \(q\)-function theory

Theorem 2.4. Let f ∈S∗ q be of the form (1.1) and µ be any complex number. Then |a3 −µa2 2| ≤max (2(1 −2µ)  ln q q −1 2 + 2  ln q q2 −1  , 2  ln q q2 −1 )

Theorem 2.5. Let f ∈S∗ q be of the form (1.1). Then |H2(2)| = |a2a4 −a2 3| ≤4  ln q q2 −1 2 . Equality occurs for the function F2(z) defined in (2.2).

Theorem 1].

Theorem 3.1]. Now we present the Herglotz representation of functions belonging to the class Cq(α):

Theorem 2.8. Let f ∈A. Then f ∈Cq(α), 0 ≤α < 1, if and only if there exists a probability measure µ supported on the unit circle such that z(Dqf)′(z) (Dqf)(z) = Z |σ|=1 σzF ′ q,α(σz)dµ(σ) where (2.3) Fq,α(z) = ∞ X n=1

Theorem 2.10. Let (2.4) Eq(z) := Iq{exp[Fq,α(z)]} = z + ∞ X n=2  1 −q 1 −qn  cnzn where cn is the n-th coefficient of the function z exp[Fq,α(z)]. Then Eq ∈Cq(α), 0 ≤α < 1. Moreover, if f(z) = z + P∞ n=2 anzn ∈Cq(α), then |an| ≤((1 −q)/(1 −qn))cn with equality holding for all n if and only if f is a rotation of Eq.

Proposition 3.1. Let f ∈Cq(α), 0 ≤α < 1. Then there exists a unique function g ∈S∗ q (α), 0 ≤α < 1, such that (3.1) g(z) = z(Dqf)(z) holds. Similarly, for a given function g ∈S∗ q (α) there exists a unique function f ∈Cq(α) satisfying (3.1).

Theorem 3.2. Let f ∈A. Then f ∈Cq(α), 0 ≤α < 1, if and only if q(Dqf)(qz) (Dqf)(z) −αq ≤1 −α, z ∈D.

Corollary 3.3. The class Cq(α) satisfies the inclusion relation \ q<p<1 Cp(α) ⊆Cq(α) and \ 0<q<1 Cq(α) = C(α).

Lemma 3.4. [1, 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 3.5. If h ∈B0 q then the infinite product Q∞ n=0{((1 −α)h(zqn) + αq)/q} converges uniformly on compact subsets of D to a nonzero function in H(D) with no zeros. Further- more, the function f satisfying the relation (3.2) z(Dqf)(z) = z Q∞ n=0{((1 −α)h(zqn) + αq)/q} belongs to Cq(α) and h(z) =  q(Dqf)(qz) (Dqf)(z) −αq 

Lemma 3.6. [9, Lemma 2.4] A function g ∈B0 q if and only if it has the representation (3.4) g(z) = exp{(ln q)p(z)}, where p(z) belongs to the class P.

Theorem 3.7. The mapping ρ : Cq(α) →B0 q defined by ρ(f)(z) =  q(Dqf)(qz) (Dqf)(z) −αq  /(1 −α) is a bijection.

Lemma 4.1. [9, Theorem 1.13] The mapping ρ : S∗ q →B0 q defined by ρ(f)(z) = f(qz) f(z) is a bijection.

Lemma 4.2. [9, Theorem 1.15] Let f ∈A. Then f ∈S∗ q 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(σz)dµ(σ) where (4.1) Fq(z) = ∞ X

Lemma 4.3. [15, pp. 254-256] Let the function p ∈P and be given by the power series (3.3). Then 2p2 = p2 1 + x(4 −p2 1), 4p3 = p3 1 + 2(4 −p2 1)p1x −p1(4 −p2 1)x2 + 2(4 −p2 1)(1 −|x|2)z, for some x and z satisfying |x| ≤1, |z| ≤1, and p1 ∈[0, 2].

Lemma 4.4. [17, Lemma 1] Let the function p ∈P and be given by the power series (3.3). Then for any real number λ, |p2 −λp2 1| ≤2 max{1, |2λ −1|} and the result is sharp.

Theorem 2.8, when the measure has a unit mass, it is clear that Eq ∈Cq(α). Let Eq(z) = z + P∞ n=2 bnzn. From this representation of Eq and the definition of Dqf, we get (4.7) z(DqEq)(z) = z + ∞ X n=2 bn(1 −qn)/(1 −q)zn.