Convex combination of analytic functions

Abstract Radii of convexity, starlikeness, lemniscate starlikeness and close-to-convexity are determined for the convex combination of the identity map and a normalized convex function F given by f(z) = α z+(1−α)F(z).


Introduction
Let A be the class of analytic functions f defined on the unit disc D D fz 2 C W jzj < 1g, and normalized by  [1], consists of f 2 A satisfying j .zf 0 .z/=f .z// 2 1j < 1 for all z 2 D, or, equivalently, if zf 0 .z/=f .z/ lies in the region bounded by the right-half of the lemniscate of Bernoulli given by jw 2 1j < 1. For recent investigation on the class SL, see [1][2][3][4][5]. Another class of our interest is the class Mˇ,ˇ> 1, consisting of f 2 A satisfying Re .zf 0 .z/=f .z// <ˇfor all z 2 D. The class Mˇwas investigated by Uralegaddi et al. [6], while its subclass was investigated by Owa and Srivastava [7]. Related radius problem for this class can be found in [8] and [9].
Properties of linear combination, in particular, convex combination of functions belonging to various classes of functions were initially investigated by Rahmanov in 1952 and1953 [10, 11]. The survey article of Campbell [12] provides several results concerning various combination of univalent functions as well as of locally univalent functions. Convex combination of univalent functions and the identity function were investigated by several authors (see Merkes [13] and references therein as well as [14]); in particular, Merkes [13] proved some results related to the present investigation. Obradovic and Nunokowa [15] investigated functions f 2 A satisfying the following condition If the function f is the convex combination f .z/ D˛z C .1 ˛/F .z/, then the condition (1) is equivalent to the conditions that F 2 CV. If two subclasses G and F of A are given, the G-radius of F, denoted by R G .F/, is the largest number R such that f .rz/=r 2 G for 0 < r < R, and for all f 2 F. Whenever G is characterized by possesing a geometric property P , the number R is also referred to as the radius of property P for the class F. In this paper, we investigate radius problem for functions f satisfying the condition (1) to belong to one of the classes introduced above. We also prove the correct results corresponding to [15, Theorem 1(a) and Theorem 2(a), p. 100] that f 2 CV if f 2 A satisfies the condition (1) for some 0 Ä˛Ä .12 p 2 15/=9. Their result is correct only when˛D 0. Unlike the radii problems associated with starlikeness and convexity, where a central feature is the estimate for the real part of the expressions zf 0 .z/=f .z/ or 1 C zf 00 .z/=f 0 .z/ respectively, the SL-radius problems are tackled by first finding the disc that contains the values of zf 0 .z/=f .z/ or 1 C zf 00 .z/=f 0 .z/. The techniques used in this paper are earlier used for the class of uniformly convex functions investigated in [16][17][18][19][20][21][22][23][24][25][26].
For two analytic functions f; g 2 A with f .z/ D z C P 1 nD2 a n z n and g.z/ D z C P 1 nD2 b n z n , their convolution or Hadamard product, denoted by f g, is defined by .f g/.z/ WD z C P 1 nD2 a n b n z n . We need the following results.   If r a is given by then fw W jw aj < r a g Â fw W jw 2 1j < 1g.

Radii problems associated with convex combinations
For functions satisfying the condition (1), we determine, in the first part of the following theorem, the range ofs o that the function is starlike of orderˇwhile the other parts of the theorem provide the radius of starlikeness of orderˇ.
For a fixed˛2 OE 1; 1/, define the function f 1 W D ! C by If f satisfies the condition (1), then it follows that the function f 1 2 CV. With the function g defined by (2), the equation (3) shows that If g is starlike in the disc D , then g. z/= is starlike and hence, by Theorem 1.2, f 1 .z/ .g. z/= / is starlike or equivalently f 1 g is starlike in the disc D . In view of this, it is enough to investigate the radius of starlikeness of the function g given by (2). For the function g given by (2), we have With z D re it and x D cos t , a calculation shows that Re Therefore, g is starlike of orderˇin jzj < 1 if, for all 0 Ä r < 1 and for all x 2 OE 1; 1, we have This inequality is equivalent to for 0 Ä r < 1 and for all x 2 OE 1; 1. It follows that the derivative of function h.r; x/ with respect to x vanishes for Therefore, for 0 Ä r < 1, min jxjÄ1 h.r; x/ > 0 and so g is starlike of orderˇ. Case (ii) Let 0 Äˇ< 1=2 and .1 2ˇ/=. 3 C 2ˇ/ <˛< 0. For fixed r, the second partial derivative of h.r; x/ with respect to x is negative and so the minimum of h.r; x/ in OE 1; 1 is attained at the end points x D˙1. Using (4) and the fact that h.r; 1/ is a decreasing function of˛, min jxjÄ1 h.r; x/ > 0 for 0 Ä r < 1, and min jxjÄ1 h.r; x/ > 0. Therefore, g is starlike of orderˇ.
To prove the sharpness, consider the functions f and g given by For the function f given in (6), we have Re The function f satisfies the condition (1) and since radius is sharp for the function g, the sharpness follows. In particular, we have the following corollary. For other ranges of˛, we have the following corollary. Theorem 2.5. For 1 Ä˛< 1, the SL-radius of the class of functions f 2 A satisfying the condition (1) is given by : Proof. First we observe that j.1 ˛z/.1 z/j .1 ˛r/.1 r/; jzj D r < 1: For 0 Ä˛< 1, (7) is trivial. For 1 Ä˛< 0, the inequality (7) holds as the function where x D cos t, is a decreasing function. Since the function g given by (2) Thus g. 2 z/= 2 2 SL. As the function f satisfies (1), the function f 1 defined by (3) is convex. Hence, by Theorem For z D 2 , the function g given by (2) satisfieš Thus, the result is sharp for the function f .z/ D g.z/ D z ˛z 2 1 z : Corollary 2.6. The SL-radius of the class CV of convex functions is 2 p 2.
Theorem 2.7. For 1=3 Ä˛Ä 1=3 andˇ> 1, the Mˇ\ ST radius of the class of functions f 2 A satisfying the condition (1) is given by Proof. From the inequality (8) for the function g given by (2) The last inequality holds if 0 < r Ä 3 .˛;ˇ/. Thus g. 3 z/= 3 2 Mˇ. Since the function f satisfies (1), the function f 1 defined by (3) where r 1 2 .0; 1 is the root of the equation in r: and r 2 2 .0; 1 is the root of the equation in r: Proof. If the function f satisfies (1) with˛D 0, then the function f is convex and so 4 .0/ D 1 as claimed. Now, assume that˛¤ 0. For the function g given by (2), a calculation shows that, with x D cos t , Re Then @ @x .r; x/ D 6r˛ 1 r 2 C 4r 2 x 2 C 3r 2˛C r 4˛Á C 24r 2 x˛ 1 C r 2˛Á : A calculation shows that The result is sharp for the function f given by (6).