On logarithmic coefficients of certain starlike functions related to the vertical strip

Lemma 1.1. (see [27]) Let φ, ϕ ∈H be any convex univalent functions in ∆. If f(z) ≺φ(z) and g(z) ≺ϕ(z), then f(z) ∗g(z) ≺φ(z) ∗ϕ(z) (z ∈∆), where ”*” denotes the Hadamard product. In this paper, some subordination relations among the classes S(α, β) and M(δ) are presented. These relations are then used to obtain sharp estimates for sums involving their logarithmic coefficients. Also, the estimate of logarithmic coefficients for functions belonging to these subclasses are determined. 2. Main Results

Theorem 2.1. Let f(z) ∈A, α < 1 and β > 1. Also let Pα,β(z) be defined by (1.5). If f(z) ∈S(α, β), then (2.1) log f(z) z  ≺bPα,β(z), where (2.2) bPα,β(z) := Z z 0 Pα,β(t) −1 t

Corollary 2.1. Let f(z) ∈S(α, β). Then f(z) z ≺exp bPα,β(z) (z ∈∆), where bPα,β(z) is given by (2.2).

Theorem 2.2. For α < 1 and β > 1, the logarithmic coefficients of f ∈S(α, β) satisfy the following inequality (2.5) ∞ X n=1 |γn|2 ≤(β −α)2 4π2 π4 45 −Li4  e−2πi 1−α β−α  −Li4

Theorem 2.3. Let f ∈A belongs to the class S(α, β) and γn be the logarithmic coefficients of f. Then (2.9) |γn| ≤β −α nπ sin π(1 −α) β −α

Corollary 2.2. If f ∈S(α, β) when β →+∞, then |γn| ≤β −α nπ sin π(1 −α) β −α ≤β −α nπ × π(1 −α) β −α = 1 −α n (n ≥1). Indeed, if f ∈S∗(α) (0 ≤α < 1) and γn is the corresponding logarithmic coeffi- cients, then we have |γn| ≤(1 −α)/n for n ≥1. Next, we have the following.

Theorem 2.4. Let π/2 ≤δ < π. Also let Bδ(z) and An be defined by (1.8) and (1.10), respectively. If f(z) ∈M(δ), then log f(z) z  ≺ Z z 0 Bδ(t) t dt. Moreover, (2.10) eBδ(z) :=

Corollary 2.3. If f(z) ∈M(δ), then f(z) z ≺exp eBδ(z) (z ∈∆), where eBδ(z) is of the form (2.10).

Theorem 2.5. Let f ∈A belongs to the class M(δ) and π/2 ≤δ < π. Then the logarithmic coefficients of f satisfy the inequality (2.11) ∞ X n=1 |γn|2 ≤ 1 16 sin2 δ π4 45 −Li4 e−2iδ
−Li4 e2iδ
 ,

Theorem 2.6. Let π/2 ≤δ < π. If f ∈A belongs to the class M(δ), then the logarithmic coefficients of f satisfy |γn| ≤1 2n (n ≥1).