Starlike functions associated with the generalized Koebe function

In this paper we study some properties of functions f which are analytic and normalized (i.e. f(0)=0=f′(0)-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(0)=0=f'(0)-1$$\end{document}) such that satisfy the following subordination relation zf′(z)f(z)-1≺z(1-pz)(1-qz),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left( \frac{zf'(z)}{f(z)}-1\right) \prec \frac{z}{(1-pz)(1-qz)}, \end{aligned}$$\end{document}where (p,q)∈[-1,1]×[-1,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p,q) \in [-1,1] \times [-1,1]$$\end{document}. These types of functions are starlike related to the generalized Koebe function. Some of the features are: radius of starlikeness of order γ∈[0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in [0,1)$$\end{document}, image of f{z:|z|<r}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\left( \{z:|z|<r\}\right) $$\end{document} where r∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\in (0,1)$$\end{document}, radius of convexity, estimation of initial and logarithmic coefficients, and Fekete–Szegö problem.

It is easy to check that each of the results cited above is true with a wider assumption ( p, q) ∈ [−1, 1] × [−1, 1].
In [8] were given bounds of minimum and maximum of the real part of function k p,q . We quote them in the following lemma.
we have the class M(β) which was introduced and investigated by Uralegaddi et al. [35] as follows Table 1 shows more details about some another subclasses of the starlike functions with different choices for ϕ.
We remark that all of the above special cases for ϕ are univalent in . But in 2011, Dziok et al. [5] defined the class S * F related to the non-univalent function p(z) which includes of all functions f ∈ A so that satisfy the following subordination relation The function p(t) is related to the Fibonacci numbers and maps the open unit disc onto a shell-like domain in the right-half plane. Motivated by the above defined classes, we introduce a new subclass of the starlike functions associated with the generalized Koebe function k p,q which is defined in (1.2). We denote this subclass by S * k ( p, q) which is defined as follows.
With a simple calculation, we see that the function (1.5) belongs to the class S * k ( p, q). Since the function f p,q is not univalent in (see Fig.  1), we conclude that the members of the class S * k ( p, q) may not be univalent in the Kuroki and Owa [18] Mendiratta et al.  whole disc . Thus it will be interesting to find the radius of univalency of functions f ∈ S * k ( p, q). Using the concept of subordination and univalency of the function k p,q (z) and also by suitable choices for p and q, we describe some geometric properties of functions f belonging to the class S * k ( p, q).

Remark 1.1
Let k p,q be given by (1.2). Then we have: (1) Suppose that p = q = 0. If f ∈ S * k (0, 0), i.e. f satisfies the following subordination relation This means that if f satisfies the above subordination relation (1.6), then it belongs to the class S(0, 2), where the class S(γ , β) (0 ≤ γ < 1, β > 1) was recently introduced by Kuroki and Owa [18]. 2), then we have the function F q (z) = z 1−q 2 z 2 . The function F q (z) was studied in [27,28]. The function F q (z) is a starlike univalent when q 2 < 1. Also F q ( ) = D(q), where It should be noted that the curve is the Booth lemniscate of elliptic type (see [27]). In the case |q| = 1, the function F q (z) becomes the function G(z) := z/(1 − z 2 ) and thus G( The function class that satisfy the last two-sided inequality was introduced by Kargar et al. [10], and studied in [1,12,13]. (5) If we take p = 0 and q = 0 in (1.2), then we get Thus by Lemma 1.1 we have If the function f ∈ A belongs to the class S * k ( p, q), then (8) Assume that p = q and 0 < p, q < 1. If the function f ∈ A belongs to the class S * k ( p, q), then (9) Let p = q and −1 < p, q < 0. If the function f ∈ A belongs to the class S * k ( p, q), then The structure of the paper is as follows. In Sect. 2 we obtain some radius problems for the class S * k ( p, q). In Sect. 3 we estimate the initial coefficients and logarithmic coefficients of the function f of the form (1.1) belonging to the class S * k ( p, q).

The radius of starlikeness and convexity
The first result of this section is contained in the following theorem.
The result is sharp.
Then from the definition of the class we have where k p,q (z) is defined by (1.2). Therefore by the subordination principle there exists a Schwarz function ω : → with ω(0) = 0 and |ω(z)| < 1 such that and consequently: .
After application of the Schwarz lemma we have so under assumptions of theorem we have h (r ) < 0 for r ∈ [0, 1]. From this we find that h(r ) is a strictly decreasing function on the interval [0, 1] and it decreases from Therefore we conclude that there is only one root of the equation h(r ) = γ in (0, 1). We can write this equation in the following equivalent form: Denote the polynomial in (2.3) by Q(r ). In the case when p or q are zero, the equation respectively. It is easy to see that in this both cases solutions are in the interval (0, 1).
Assume now that pq = 0. Then Q is a quadratic polynomial with determinant of the form and we can see that this determinant is positive for all p, q; pq = 0. In consequence, there are two roots of Q: From this it follows that the roots r 1 , r 2 both are positive numbers. Let us recall that the equation h(r ) = γ has strictly one solution in (0, 1) so the equation Q(r ) = 0 has. From this it follows that this solution is r 1 . Therefore f is starlike of order γ in the disc |z| < r < r s ( p, q, γ ) where r s ( p, q, γ ) is given by (2.1).
For the sharpness consider the function f p,q given by (1.5). It is easy to see that With the same argument as above we get the result. Here the proof ends.
Putting γ = 0 in the previous theorem we obtain the following result: The result is sharp.

4)
then each function f ∈ S * k ( p, q) maps a disc |z| < r onto a starlike domain. The result is sharp.
After repeating the same reasoning as in the proof of Theorem 2.1 we have that for f ∈ S * k ( p, q) the following condition holds Moreover for |z| < r we obtain It is easy to observe that under our assumptions, the function l( p, q) has positive values. In conclusion we obtain the thesis. The function f p,q shows that the result is sharp concluding the proof. Note that, for the homography q( p) = q = r p+r −1 r 2 p−r its vertical asymptote and horizontal asymptote are given by the equations p = 1 r and q = 1 r , respectively. Moreover, zero of this homography is the point p = 1 r − 1 < 1 r and 0 < q(0) = 1 r − 1 < 1 r . The suitable set of the ( p, q) is bounded by one of the branches of hyperbola and by p-axis and q-axis (domain D 1 , Fig. 2).
Therefore taking into account the symmetry of the set D(r ) we conclude that it has a form as in Fig. 2.  , q), it follows that there exists a Schwarz function w such that (2.2) holds true. A logarithmic differentiation of (2.2) gives

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
The Schwarz-Pick lemma (see [24]) states that for a Schwarz function w the following sharp estimate holds Also if w is a Schwarz function then |w(z)| ≤ |z| (cf. [4]). According to what came above and using definition of convexity, it follows from (2.5) that It is a simple exercise that F( p, q, r ) > 0 if and only if 0 < r ≤ δ where δ is the smallest positive root of This is the end of proof. F( p, q, r ) be defined as (2.6). It is easy to check that F(1/2, 1/2, r ) = 0 has two real roots as follows because F(0, 0, r ) has three real roots r 3 ≈ 0.55496, r 4 ≈ −0.80194 and r 5 ≈ 2.2470.

On coefficients of f ∈ S * k (p, q)
Following, we shall estimate the initial coefficients and Fekete-Szegö problem for the function f of the form (1.1) belonging to the class S * k ( p, q). The following lemmas will be useful. (3.1) Then |c 1 | ≤ 1 and |c n | ≤ 1 − |c 1 | 2 (n = 2, 3, . . .). [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:

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 The result is sharp for the functions w(z) = z 2 or w(z) = z. and where H (ρ, τ ) is of the form as in Lemma 3.2. Further All inequalities are sharp.
Proof Let f ∈ S * k ( p, q). Then by Definition 1.1 and subordination principle, there exists a Schwarz function w of the form (3.1) with |w(z)| < 1 such that where k p,q is defined in (1.2). Furthermore, since f has the form (1.1), it is easy to see that Also, using (1.2) and (3.1), we get where and 3 p 2 , p = q, (3.9) are defined in (1.2). Comparing (3.6) and (3.7), gives us a 2 = c 1 (3.10) The inequality |a 2 | ≤ 1 follows directly from Lemma 3.1 and (3.10) with sharpness for the function f p,q given by (1.5). From Lemma 3.1 we have Therefore using (3.11) and the last estimate give that |a 3 | ≤ 1 2 (1 + |A 2 |). Now by (3.8) and since ( p, q) ∈ [−1, 1] × [−1, 1], we get the desired inequality (3.4). Now we shall find the estimation of the fourth coefficient. For this we first use (3.8) and (3.9) in (3.12) to obtain By setting and by applying Lemma 3.2, we can write where the function H is defined in (3.2). Thus the result is established. Now let μ be a complex number. From (3.10) and (3.11) we get Therefore using Lemma 3.3 we obtain It easy to see that equalities in (3.3)-(3.4) occur for the function f p,q defined by (1.5). Sharpness the third inequality (3.5) also follows as an application of Lemma 3.2. This completes the proof.
At the end of this paper we discuss the logarithmic coefficients γ n := γ n ( f ) of the functions f belonging to the class S * k ( p, q). We recall that the logarithmic coefficients γ n of f ∈ S are defined with the following series expansion: The logarithmic coefficients have an important role in Geometric Function Theory. We remark that Kayumov [14] by use of these coefficients and under an additional condition solved the Brennan conjecture for conformal mappings or before de Branges by use of this concept, was able to prove the famous Bieberbach's conjecture [2]. We recall that the logarithmic coefficients γ n of every function f (z) = z+ ∞ n=2 a n z n ∈ S satisfy the inequalities The sharp estimate of |γ n | when n ≥ 3 and f ∈ S is still open.
In the sequel, we derive some inequalities involving the logarithmic coefficients in the class S * k ( p, q). Theorem 3.2 If a function f ∈ A belongs to the class S * k ( p, q) and γ n is the logarithmic coefficient of f , then the following sharp inequality holds where H (ρ, τ ) is of the form as in Lemma 3.2, and A 2 and A 3 are defined in (3.8) and (3.9), respectively. All inequalities are sharp.
For the sharpness we consider the function f p,q defined by (1.5). A simple check gives that ∞ n=1 2γ n ( f p,q )z n = log f p,q (z) Comparison of the corresponding coefficients and an application of Lemma 3.2 show the result is sharp, therefore the proof is completed.
For the next result we need the following theorem. By using this theorem we give the sharp inequality for sums involving logarithmic coefficients.
We consider the following cases: Case 1. Let q = 0 = p. Then we have where Li 2 denotes the dilogarithm function. For the sharpness, it is enough to consider the function K p,q (z) := z exp(K p,q (z)), where K p,q is defined by (3.15). It is easily seen that K p,q (z) ∈ S * k ( p, q) and γ n ( K p,q (z)) = A n 2n .
Thus the proof is complete.