Radii problems for some classes of analytic functions associated with Legendre polynomials of odd degree

The aim of the present paper is to study radii problems for two general classes including various known subclasses of analytic functions associated with the normalized form of Legendre polynomials of odd degree. We also obtain some special cases of the main results presented here with some useful examples.


Introduction and preliminaries
Let U(r) := {z ∈ C : |z| < r} be the disk in the complex plane C centered at the origin, with radius r > 0, and denote by U := U(1) the unit disk. We denote by A the class of analytic functions in the unit disk U normalized by f (0) = f (0) -1 = 0, and let S be the subclass of A consisting of univalent functions.
We denote by S * (α) the subclass of A consisting of functions which are starlike of order α in U, that is, Also, let us denote by S * (α) the subclass of A consisting of functions which are strongly starlike of order α in U, that is, are called the radius of starlikeness of order α and the radius of strong starlikeness of order α of the function f , respectively. In particular, r * (f ) := r * 0 (f ) = r * 1 (f ) is called the radius of starlikeness of the function f .
It is well known that the Rodrigues formula implies that the Legendre polynomials of odd degree have only real roots, and the roots of P 2n-1 (z) are 0 = z 0 < z 1 < · · · < z n-1 and -z 1 , -z 2 , . . . , -z n-1 , while the product representation of the polynomial P 2n-1 is Bulut and Engel [7] have obtained the radius of starlikeness, convexity, and uniform convexity (see [10,12,13]) of the normalized form of the Legendre polynomial of odd degree. In the recent years, several authors determined the radius of starlikeness, convexity, and uniform convexity for some special functions, that is a relative new direction in the geometric function theory (see, for example, [1][2][3][4][5][6]).
In the present paper we obtain the radius of strong starlikeness and other related radius of the normalized form of Legendre polynomials of odd degree, and the technique of the proofs used in our paper is similar to that of several papers [1,[3][4][5][6]. Further, our results are well supported by some examples.
In order to prove our main results, we require the following lemmas.

Main results
Using the first of the above lemmas, we obtain the k -UCST (α) radius of P 2n-1 as follows.
Let ω : I → R, where I is the open interval (0, z 1 ) which is subset of R be the function defined by Since lim r 0 ω(r) = 1, lim r z 1 ω(r) = -∞, and the function ω is continuous, it follows that the equation ω(r) = 0 has at least a root in (0, z 1 ). Thus, if r 1 is the smallest positive root of the equation ω(r) = 0, then we have for |z| < r 1 , and inf |z|<r 1 It follows that r k-UCST α (P 2n-1 ) = r 1 is the radius of k -UCST (α) of the normalized Legendre polynomial P 2n-1 , and hence this completes our proof.
Choosing k = 0 in Theorem 1, we obtain the next result which was given by Bulut and Engel for β = 0 in [7, Theorem 2.2].

Example 1 For n = 2 we have
Like we see from Fig. 1(a) the domain P 3 (U) is not convex; moreover, the function P 3 is not univalent in U. From Corollary 1 it follows that the radius of convexity of P 3 is r c (P 3 ) = 1/ √ 15 0.2581988897 . . . , where 1/ √ 15 denotes the smallest positive root of the equation rP 3 (r) + P 3 (r) = -15r 2 + 1 = 0.
According to the above result, the domain P 3 (U(r c (P 3 ))) shown in Fig. 1(b) is convex. Letting α = 1 in Theorem 1, we obtain the next special case.

Corollary 2
The radius of k-uniformly convexity of P 2n-1 is r ucv (P 2n-1 ) = r 3 , where r 3 denotes the smallest positive root of the equation Setting k = 1 in Corollary 2, we obtain the following result which was given by Bulut and Engel in [7,Theorem 2.3]. According to the above result, the domain P 5 (U(r uc (P 5 ))) is uniformly convex, and it is plotted in Fig. 2(b).
Letting α = 0 in Theorem 1, we deduce the next result. Letting k = 1 in Corollary 3, we obtain the following special case.
In the following theorem we obtain the radius of strong starlikeness of order α of P 2n-1 .
Using the above inequalities, from relation (5) we get for |z| ≤ r < z 1 . Denoting we see that Im a = 0, and from Lemma 2 and the above inequality it follows that the disc |w -a| ≤ R a is contained in the sector | arg w| ≤ πα/2, that is, if we assume that the inequality The above inequality implies that ψ(r) ≤ 0 for r ∈ (0, z 1 ). Also, we have lim r 0 ψ(r) = -sin πα 2 < 0 and lim r z 1 ψ(r) = +∞. On the other the hand, we have ψ (r) ≥ 0 for z ∈ (0, z 1 ). It follows that the equation ψ(r) = 0 has a unique root r * 1 in (0, z 1 ). Therefore, the radius of strong starlikeness of order α of P 2n-1 is r * α (P 2n-1 ) = r * 1 .
Letting α = 1 in the above theorem, we get the following corollary.
Remark 2 All the figures inserted in this article have been obtained using MAPLE™ software.