Starlike functions associated with the generalized Koebe function

Lemma 1.1 Let (p, q) ∈[−1, 1] × [−1, 1] and |pq| ̸= 1. The values of max 0≤t≤2π Re  kp,q(eit)  and min 0≤t≤2π Re  kp,q(eit)  are the following min 0≤t≤2π Re

Theorem 2.1 Let (p, q) ∈[−1, 1] × [−1, 1] and γ ∈[0, 1). If f ∈S∗ k (p, q), then f is starlike of order γ in the disc |z| < rs(p, q, γ ) where rs(p, q, γ ) = ⎧ ⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎩ 1−γ 1+(1−γ )|q| i f p = 0, 1−γ 1+(1−γ )|p| i f q = 0,

Corollary 2.1 Let (p, q) ∈[−1, 1] × [−1, 1] and γ ∈[0, 1). If f ∈S∗ k (p, q), then f is starlike univalent in the disc |z| < rs(p, q) where rs(p, q) = ⎧ ⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎩ 1 1+|q| i f p = 0, 1 1+|p| i f q = 0,

Theorem 2.2 Let the number r ∈(0, 1] be given and (p, q) ∈[−1, 1] × [−1, 1]. If |q| < r|p| + r −1 r2|p| −r , (2.4) then each function f ∈S∗ k (p, q) maps a disc |z| < r onto a starlike domain. The result is sharp.

Theorem 2.2. For given r ∈(0, 1], let D(r) by the set of solutions of the inequality (2.4). Observe that due to the form of this inequality, D(r) must be symmetrical about both axes. Let us find its part lying in the first quadrant of the coordinate system. If p ≥0 and q ≥0 then (2.4) reduces to the condition q < rp + r −1 r2 p −r .

Corollary 2.2 Let 0 < r ≤3− √ 5 2 = 0.381966 . . . be given. Then for each function f ∈S∗ k (p, q) the set f (|z| < r) is a starlike domain.

Theorem 2.3 Let a function f ∈A belongs to the class S∗ k (p, q). Then f is convex univalent in the disk |z| < δ where δ is the smallest positive root of equation 1 − r (1 −|p|r)(1 −|q|r) −  2|p||q|r + 1 + |p| + |q| (1 −|p|r)(1 −|q|r) −r + |p| 1 −|p|r + |q| 1 −|q|r  r

Lemma 3.1 (Nehari [24, p. 172]) Let w be a Schwarz function of the form w(z) = ∞  n=1 cnzn (z ∈). (3.1) Then |c1| ≤1 and |cn| ≤1 −|c1|2 (n = 2, 3, . . .).

Lemma 3.2 (Prokhorov and Szynal [29]) If w is a Schwarz function of the form (3.1), then for any complex numbers ρ and τ the following sharp estimate holds: |c3 + ρc1c2 + τc3 1| ≤H(ρ, τ),

Lemma 3.3 (Keogh and Merkes [15]) Let w be a Schwarz function of the form (3.1). Then for any complex number μ we have |c2 −μc2 1| ≤max{1, |μ|}. The result is sharp for the functions w(z) = z2 or w(z) = z.

Theorem 3.1 Let f of the form (1.1) belong to the class S∗ k (p, q) where (p, q) ∈ [−1, 1] × [−1, 1]. Then the following inequalities for the coefficients of f hold |a2| ≤1, (3.3) |a3| ≤ ⎧ ⎨ ⎩ 1 2(1 + |p| + |q|), p ̸= q; 1 2(1 + 2|p|), p = q (3.4)

Theorem 3.2 If a function f ∈A belongs to the class S∗ k (p, q) and γn is the loga- rithmic coefficient of f , then the following sharp inequality holds |γ1| ≤1 2, |γ2| ≤ ⎧ ⎨ ⎩ 1 4(|p| + |q|), p ̸= q&|p + q| ≥1; 1 2|p|, p = q&|p| ≥1/2 and

Theorem 3.3 Let the function f ∈A belong to the class S∗ k (p, q) and kp,q(z) be defined by (1.2). Then log  f (z) z  ≺  z 0 kp,q(t) t dt.

Theorem 3.4 Let the f ∈A belong to the class S∗ k (p, q). Then the logarithmic coef- ficients of f satisfy the following sharp inequality ∞  n=1 |γn|2 ≤ ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ 1 4|p−q|2 ∞ n=1 1

Theorem 3.2.