Radius Constants for Analytic Functions with Fixed Second Coefficient

Abstract

Let 𝑓(𝑧)=𝑧+∑^∞_𝑛=2𝑎_𝑛𝑧^𝑛 be analytic in the unit disk with the second coefficient 𝑎₂ satisfying |𝑎₂|=2𝑏, 0≤𝑏≤1. Sharp radius of Janowski starlikeness is obtained for functions 𝑓 whose 𝑛th coefficient satisfies |𝑎_𝑛|≤𝑐𝑛+𝑑(𝑐,𝑑≥0) or |𝑎_𝑛|≤𝑐/𝑛(𝑐>0and𝑛≥3). Other radius constants are also obtained for these functions, and connections with earlier results are made.

1. Introduction

Let 𝒜 denote the class of analytic functions 𝑓 defined in the open unit disk 𝔻∶={𝑧∈ℂ∶|𝑧|<1}, normalized by 𝑓(0)=0=𝑓^′(0)−1, and let 𝒮 denote its subclass consisting of univalent functions. If 𝑓(𝑧)=𝑧+∑^∞_𝑛=2𝑎_𝑛𝑧^𝑛∈𝒮, de Branges [1] obtained the sharp coefficient bound that |𝑎_𝑛|≤𝑛(𝑛≥2). However, the inequality |𝑎_𝑛|≤𝑛, 𝑛≥2, is not sufficient for 𝑓 to be univalent; for example, 𝑓(𝑧)=𝑧+2𝑧² is clearly not a member of 𝒮.

Several subclasses of 𝒮 possess a similar coefficient bound. For instance, the 𝑛th coefficients of starlike functions, convex functions in the direction of imaginary axis, and close-to-convex functions satisfy |𝑎_𝑛|≤𝑛(𝑛≥2) [2–4]. Other examples include functions which are convex, starlike of order 1/2, and starlike with respect to symmetric points. The 𝑛th coefficients of these functions satisfy |𝑎_𝑛|≤1(𝑛≥2) [5–7]. The 𝑛th coefficient of close-to-convex functions with argument 𝛽 satisfies |𝑎_𝑛|≤1+(𝑛−1)cos𝛽 [8], and the coefficients of uniformly starlike functions are bounded by 2/𝑛 [9], while |𝑎_𝑛|≤1/𝑛 [10] for uniformly convex functions. Simple examples show that these bounds are not sufficient to characterize the geometric properties of the classes of functions.

In the sequel, we will assume that 𝑓∈𝒜 has the Taylor expansion of the form 𝑓(𝑧)=𝑧+∑^∞_𝑛=2𝑎_𝑛𝑧^𝑛. Gavrilov [11] showed that the radius of univalence for functions 𝑓∈𝒜 satisfying |𝑎_𝑛|≤𝑛(𝑛≥2) is the real root 𝑟₀≃0.164 of the equation 2(1−𝑟)³−(1+𝑟)=0, and the result is sharp for 𝑓(𝑧)=2𝑧−𝑧/(1−𝑧)². Gavrilov also proved that the radius of univalence for functions 𝑓∈𝒜 satisfying the coefficient bound |𝑎_𝑛|≤𝑀(𝑛≥2) is 1−𝑀/(1+𝑀). The condition |𝑎_𝑛|≤𝑀 clearly holds for functions 𝑓∈𝒜 satisfying |𝑓(𝑧)|≤𝑀, and for these functions, Landau [12] proved that the radius of univalence is 𝑀−𝑀²−1. In fact, Yamashita [13] showed that the radius of univalence obtained by Gavrilov [11] is also the radius of starlikeness for functions 𝑓∈𝒜 satisfying |𝑎_𝑛|≤𝑛 or |𝑎_𝑛|≤𝑀. Additionally, Yamashita [13] determined that the radius of convexity for functions 𝑓∈𝒜 satisfying |𝑎_𝑛|≤𝑛 is the real root 𝑟₀≃0.090 of the equation 2(1−𝑟)⁴−(1+4𝑟+𝑟²)=0, while the radius of convexity for functions 𝑓∈𝒜 satisfying |𝑎_𝑛|≤𝑀 is the real root of (𝑀+1)(1−𝑟)³−𝑀(1+𝑟)=0. (1) Recently, Kalaj et al. [14] obtained the radii of univalence, starlikeness, and convexity for harmonic mappings satisfying certain coefficient inequalities.

For two analytic functions 𝑓 and 𝑔, the function 𝑓 is subordinate to 𝑔, denoted by 𝑓≺𝑔, if there is an analytic self-map 𝑤 of 𝔻 with 𝑤(0)=0 satisfying 𝑓(𝑧)=𝑔(𝑤(𝑧)). If 𝑔 is univalent, then 𝑓≺𝑔 is equivalent to 𝑓(0)=𝑔(0) and 𝑓(𝔻)⊆𝑔(𝔻).

For 𝛽∈ℝ⧵{1},  𝛼≥0, the class ℒ(𝛼,𝛽) consists of functions 𝑓∈𝒜 satisfying 𝛼𝑧²𝑓^′′(𝑧)𝑓(𝑧)+𝑧𝑓^′(𝑧)𝑓(𝑧)≺1+1−2𝛽𝑧.1−𝑧 (2) Denote by ℒ₀(𝛼,𝛽) its subclass consisting of functions 𝑓∈𝒜 satisfying 𝛼𝑧²𝑓^′′(𝑧)𝑓(𝑧)+𝑧𝑓^′(𝑧)𝑓(𝑧)−1≤1−𝛽𝛽∈ℝ⧵{1},𝛼≥0. (3) These classes were investigated in [15–24].

For 𝛽<1, the class ℒ(0,𝛽) is the class of starlike functions of order 𝛽, while, for the case 𝛽>1, the class was studied in [25–28].

The class 𝒮𝒯[𝐴,𝐵] of Janowski starlike functions [29] consists of 𝑓∈𝒜 satisfying the subordination 𝑧𝑓^′(𝑧)𝑓(𝑧)≺1+𝐴𝑧().1+𝐵𝑧−1≤𝐵<𝐴≤1 (4) Certain well-known subclasses of starlike functions are special cases of 𝒮𝒯[𝐴,𝐵] for appropriate choices of the parameters 𝐴 and 𝐵. For example, for 0≤𝛽<1, 𝒮𝒯(𝛽)∶=𝒮𝒯[1−2𝛽,−1] is the familiar class of starlike functions of order 𝛽. Denote by 𝒮𝒯_𝛽 the class 𝒮𝒯_𝛽∶=ℒ₀(0,𝛽)=𝒮𝒯[1−𝛽,0]. Janowski [29] obtained the sharp radius of convexity for 𝒮𝒯[𝐴,𝐵].

This paper studies the class 𝒜_𝑏 consisting of functions 𝑓(𝑧)=𝑧+∑^∞_𝑛=2𝑎_𝑛𝑧^𝑛,  (|𝑎₂|=2𝑏,0≤𝑏≤1), in the disk 𝔻. The subclass of univalent functions in 𝒜_𝑏 have been studied in [30–33]. In [33], Ravichandran obtained sharp radii of starlikeness and convexity of order 𝛼 for functions 𝑓∈𝒜_𝑏 satisfying |𝑎_𝑛|≤𝑛 or |𝑎_𝑛|≤𝑀, 𝑛≥3. The author also obtained the radius of uniform convexity and parabolic starlikeness for functions 𝑓∈𝒜_𝑏 satisfying |𝑎_𝑛|≤𝑛, 𝑛≥3.

This paper finds radius constants for functions 𝑓(𝑧)=𝑧+∑^∞_𝑛=2𝑎_𝑛𝑧^𝑛∈𝒜_𝑏 satisfying either |𝑎_𝑛|≤𝑐𝑛+𝑑 (𝑐,𝑑≥0) or |𝑎_𝑛|≤𝑐/𝑛  (𝑐>0,𝑛≥3). In the next section, sharp ℒ(𝛼,𝛽)-radius and 𝒮𝒯[𝐴,𝐵]-radius are derived for these classes. Several known radius constants are shown to be special cases of the results obtained.

2. Radius Constants

A sufficient condition for functions 𝑓∈𝒜 to belong to the class ℒ(𝛼,𝛽) is given in the following lemma.

Lemma 1 (see [24, 34]). Let 𝛽∈ℝ⧵{1} and 𝛼≥0. If 𝑓(𝑧)=𝑧+∑^∞_𝑛=2𝑎_𝑛𝑧^𝑛∈𝒜 satisfies the inequality ^∞_𝑛=2𝛼𝑛²+(1−𝛼)𝑛−𝛽𝑎_𝑛≤1−𝛽, (5) then 𝑓∈ℒ(𝛼,𝛽).

Making use of this lemma, the sharp ℒ(𝛼,𝛽)-radius is obtained for 𝑓∈𝒜_𝑏 satisfying the coefficient inequality |𝑎_𝑛|≤𝑐𝑛+𝑑.

Theorem 2. Let 𝛽∈ℝ⧵{1}, 6𝛼+3−𝛽≥0, and 𝛼≥0. The ℒ(𝛼,𝛽)-radius for 𝑓(𝑧)=𝑧+∑^∞_𝑛=2𝑎_𝑛𝑧^𝑛∈𝒜_𝑏 satisfying the coefficient inequality |𝑎_𝑛|≤𝑐𝑛+𝑑, 𝑐,𝑑≥0, 𝑛≥3, is the real root in (0,1) of the equation ()++(2())𝑟()𝑐+𝑑1−𝛽1−𝛽2𝛼+2−𝛽𝑐−𝑏+𝑑1−𝑟⁴=𝑐𝛼1+4𝑟+𝑟²+((1−𝛼)𝑐+𝛼𝑑)1−𝑟²+(1−𝛼)𝑑−𝛽𝑐(1−𝑟)²−𝛽𝑑(1−𝑟)³. (6) For 𝛽<1, this number is also the ℒ₀(𝛼,𝛽)-radius of 𝑓∈𝒜_𝑏. The results are sharp.

Proof. The number 𝑟₀ is the ℒ(𝛼,𝛽)-radius for 𝑓∈𝒜_𝑏 if and only if 𝑓(𝑟₀𝑧)/𝑟₀∈ℒ(𝛼,𝛽). Therefore, by Lemma 1, it is sufficient to verify the inequality ^∞_𝑛=2𝛼𝑛²+(1−𝛼)𝑛−𝛽𝑎_𝑛𝑟₀^𝑛−1≤1−𝛽, (7) where 𝑟₀ is the real root in (0,1) of (6). Using the known expansions ^∞_𝑛=3𝑟₀^𝑛−1=11−𝑟₀−1−𝑟₀,_∞𝑛=3𝑛𝑟₀^𝑛−1=11−𝑟₀²−1−2𝑟₀,_∞𝑛=3𝑛²𝑟₀^𝑛−1=1+𝑟₀1−𝑟₀³−1−4𝑟₀,_∞𝑛=3𝑛³𝑟₀^𝑛−1=1+4𝑟₀+𝑟²₀1−𝑟₀⁴−1−8𝑟₀ (8)(9)(10)(11) leads to ^∞_𝑛=2𝛼𝑛²+(1−𝛼)𝑛−𝛽𝑎_𝑛𝑟₀^𝑛−1≤22𝛼+2−𝛽𝑏𝑟₀+^∞_𝑛=3𝛼𝑛²+(1−𝛼)𝑛−𝛽(𝑐𝑛+𝑑)𝑟₀^𝑛−1=22𝛼+2−𝛽𝑏𝑟₀+𝑐𝛼1+4𝑟₀+𝑟²₀1−𝑟₀⁴−1−8𝑟₀+((1−𝛼)𝑐+𝛼𝑑)1+𝑟₀1−𝑟₀³−1−4𝑟₀+()11−𝛼𝑑−𝛽𝑐1−𝑟₀²−1−2𝑟₀1−𝛽𝑑1−𝑟₀−1−𝑟₀=(𝑐+𝑑)𝛽−1−2𝛼+2−𝛽(2(𝑐−𝑏)+𝑑)𝑟₀+𝑐𝛼1+4𝑟₀+𝑟²₀+((1−𝛼)𝑐+𝛼𝑑)1−𝑟²₀+(1−𝛼)𝑑−𝛽𝑐1−𝑟₀²−𝛽𝑑1−𝑟₀³×1−𝑟₀⁻⁴.=1−𝛽. (12)
For 𝛽<1, consider the function 𝑓₀(𝑧)=𝑧−2𝑏𝑧²−^∞_𝑛=3(𝑐𝑛+𝑑)𝑧^𝑛=(𝑐+1)𝑧+2(𝑐−𝑏)𝑧²−𝑐𝑧(1−𝑧)²−𝑑𝑧³.1−𝑧 (13) At the root 𝑧=𝑟₀ in (0,1) of (6), 𝑓₀ satisfies 𝛼𝑧Re²𝑓₀^′′(𝑧)𝑓₀(𝑧)+𝑧𝑓^′₀(𝑧)𝑓₀(𝑧)𝑁𝑟=1−⁰𝐷𝑟₀=𝛽, (14) where 𝑁𝑟₀=−2(𝑐−𝑏)(2𝛼+1)𝑟₀+2𝑐𝑟₀(2𝛼+1)1−𝑟₀³+6𝑐𝛼𝑟²₀1−𝑟₀⁴+2𝑑𝑟²₀(3𝛼+1)1−𝑟₀+𝑑𝑟³₀(6𝛼+1)1−𝑟₀²+2𝑑𝑟⁴₀𝛼1−𝑟₀³,𝐷𝑟₀=𝑐+1+2(𝑐−𝑏)𝑟₀−𝑐1−𝑟₀²−𝑑𝑟²₀1−𝑟₀. (15) This shows that 𝑟₀ is the sharp ℒ(𝛼,𝛽)-radius for 𝑓∈𝒜_𝑏. For 𝛽<1, (14) shows that the rational expression 𝑁(𝑟₀)/𝐷(𝑟₀) is positive, and therefore the equality 𝛼𝑧²𝑓₀^′′(𝑧)𝑓₀(𝑧)+𝑧𝑓^′₀(𝑧)𝑓₀(𝑧)−1=1−𝛽 (16) holds. Thus, 𝑟₀ is the sharp ℒ₀(𝛼,𝛽)-radius for 𝑓∈𝒜_𝑏 when 𝛽<1.
For 𝛽>1, the function 𝑓₀(𝑧)=𝑧+2𝑏𝑧²+^∞_𝑛=3(𝑐𝑛+𝑑)𝑧^𝑛=(1−𝑐)𝑧+2(𝑏−𝑐)𝑧²+𝑐𝑧(1−𝑧)²+𝑑𝑧³1−𝑧 (17) demonstrates sharpness of the result. The derivation is similar to the case 𝛽<1 and is omitted.

Theorem 3. Let 𝛽∈ℝ⧵{1} and 𝛼≥0. The ℒ(𝛼,𝛽)-radius of 𝑓(𝑧)=𝑧+∑^∞_𝑛=2𝑎_𝑛𝑧^𝑛∈𝒜_𝑏 satisfying the coefficient inequality |𝑎_𝑛|≤𝑐/𝑛 for 𝑛≥3 and 𝑐>0 is the real root in (0,1) of the equation 𝑐++𝑟𝑐1−𝛽1−𝛽2𝛼+2−𝛽2−2𝑏(1−𝑟)²=𝑐𝛼+(1−𝛼)𝑐(1−𝑟)+𝛽𝑐(1−𝑟)²log(1−𝑟)𝑟. (18) For 𝛽<1, this number is also the ℒ₀(𝛼,𝛽)-radius of 𝑓∈𝒜_𝑏. The results are sharp.

Proof. By Lemma 1, 𝑟₀ is the ℒ(𝛼,𝛽)-radius of functions 𝑓∈𝒜_𝑏 when inequality (7) holds for the real root 𝑟₀ of (18) in (0,1). Using (8) and (9) together with ^∞_𝑛=3𝑟₀^𝑛−1𝑛=−log1−𝑟₀𝑟₀𝑟−1−⁰2 (19) leads to ^∞_𝑛=2𝛼𝑛²+(1−𝛼)𝑛−𝛽𝑎_𝑛𝑟₀^𝑛−1≤22𝛼+2−𝛽𝑏𝑟₀+^∞_𝑛=3𝛼𝑛²+()𝑐1−𝛼𝑛−𝛽𝑛𝑟₀^𝑛−1=22𝛼+2−𝛽𝑏𝑟₀1+𝑐𝛼1−𝑟₀²−1−2𝑟₀+()𝑐11−𝛼1−𝑟₀−1−𝑟₀−𝛽𝑐−log1−𝑟₀𝑟₀𝑟−1−⁰2=𝑐𝛽−1+2𝛼+2−𝛽𝑟₀𝑐2𝑏−2+𝑐𝛼𝑟₀+(1−𝛼)𝑐1−𝑟₀𝑟₀+𝛽𝑐1−𝑟₀²log1−𝑟₀1−𝑟₀²𝑟₀=1−𝛽. (20)
To verify sharpness for 𝛽<1, consider the function 𝑓₀(𝑧)=𝑧−2𝑏𝑧²−^∞_𝑛=3𝑐𝑛𝑧^𝑛=()𝑐1+𝑐𝑧+2−2𝑏𝑧²+𝑐log(1−𝑧). (21) At the root 𝑧=𝑟₀ in (0,1) of (18), 𝑓₀ satisfies 𝛼𝑧Re²𝑓₀^′′(𝑧)𝑓₀(𝑧)+𝑧𝑓^′₀(𝑧)𝑓₀(𝑧)−𝑐=1−2−2𝑏𝑟₀(2𝛼+1)+𝑐𝑟₀𝛼1−𝑟₀²+𝑐1−𝑟₀+𝑐log1−𝑟₀𝑟₀×()+𝑐1+𝑐2−2𝑏𝑟₀+𝑐log1−𝑟₀𝑟₀⁻¹=𝛽. (22) Thus, 𝑟₀ is the sharp ℒ(𝛼,𝛽)-radius for 𝑓∈𝒜_𝑏. For 𝛽<1, the rational expression in (22) is positive, and therefore 𝛼𝑧²𝑓₀^′′(𝑧)𝑓₀(𝑧)+𝑧𝑓^′₀(𝑧)𝑓₀(𝑧)−1=1−𝛽, (23) which shows that 𝑟₀ is the sharp ℒ₀(𝛼,𝛽)-radius for 𝑓∈𝒜_𝑏. For 𝛽>1, sharpness of the result is demonstrated by the function 𝑓₀ given by 𝑓₀(𝑧)=𝑧+2𝑏𝑧²+^∞_𝑛=3𝑐𝑛𝑧^𝑛=()𝑐1−𝑐𝑧+2𝑏−2𝑧²−𝑐log(1−𝑧). (24)

Remark 4. The results obtained above yield the following special cases. (1)For 𝛼=0, 𝛽=0, 𝑐=1, 𝑑=0, and 0≤𝑏≤1, Theorem 2 yields the radius of starlikeness obtained by Yamashita [13].(2)For 𝛼=0, 𝑐=1, and 𝑑=0, Theorem 2 reduces to Theorem  2.1 in [33, page 3]. When 𝛼=0, 𝑐=0, and 𝑑=𝑀, Theorem 2 leads to Theorem  2.5 in [33, page 5].(3)For 𝛼=0, Theorem 3 yields the radius of starlikeness of order 𝛽 for 𝑓∈𝒜_𝑏 obtained by Ravichandran [33, Theorem 2.8].

The following result of Goel and Sohi [35] will be required in our investigation of the class of Janowski starlike functions.

Lemma 5 (see [35]). Let −1≤𝐵<𝐴≤1. If 𝑓(𝑧)=𝑧+∑^∞_𝑛=2𝑎_𝑛𝑧^𝑛∈𝒜 satisfies the inequality ((1−𝐵)^∞_𝑛=2𝑛−(1−𝐴))𝑎_𝑛≤𝐴−𝐵, (25) then 𝑓∈𝒮𝒯[𝐴,𝐵].

The next result finds the sharp 𝒮𝒯[𝐴,𝐵]-radius for 𝑓∈𝒜_𝑏 satisfying the coefficient inequality |𝑎_𝑛|≤𝑐𝑛+𝑑.

Theorem 6. Let −1≤𝐵<𝐴≤1. The 𝒮𝒯[𝐴,𝐵]-radius for 𝑓(𝑧)=𝑧+∑^∞_𝑛=2𝑎_𝑛𝑧^𝑛∈𝒜_𝑏 satisfying the coefficient inequality |𝑎_𝑛|≤𝑐𝑛+𝑑, 𝑛≥3 and 𝑐,𝑑≥0, is the real root in (0,1) of the equation )−(2()−(𝑟)[(𝐴−𝐵)(𝑐+𝑑+12𝑏−2𝑐−𝑑)(1−𝐵1−𝐴))](1−𝑟³()+(𝑑()()−()𝑑()=𝑐1−𝐵)(1+𝑟1−𝐵−𝑐1−𝐴))(1−𝑟1−𝐴1−𝑟². (26) This radius is sharp.

Proof. It is evident that 𝑟₀ is the 𝒮𝒯[𝐴,𝐵]-radius of 𝑓∈𝒜_𝑏 if and only if 𝑓(𝑟₀𝑧)/𝑟₀∈𝒮𝒯[𝐴,𝐵]. Hence, by Lemma 5, it suffices to show that ((1−𝐵)^∞_𝑛=2𝑛−(1−𝐴))𝑎_𝑛𝑟₀^𝑛−1(),≤𝐴−𝐵−1≤𝐵<𝐴≤1 (27) where 𝑟₀ is the root in (0,1) of (26). From (8), (9), and (10), it follows that ((1−𝐵)^∞_𝑛=2𝑛−(1−𝐴))𝑎_𝑛𝑟₀^𝑛−1≤2(2(1−𝐵)−(1−𝐴))𝑏𝑟₀+^∞_𝑛=3((1−𝐵)𝑛−(1−𝐴))(𝑐𝑛+𝑑)𝑟₀^𝑛−1=2(2(1−𝐵)−(1−𝐴))𝑏𝑟₀+𝑐(1−𝐵)1+𝑟₀1−𝑟₀³−1−4𝑟₀+(𝑑()(11−𝐵−𝑐1−𝐴))1−𝑟₀²−1−2𝑟₀−()𝑑11−𝐴1−𝑟₀−1−𝑟₀=()+()×(2()−(𝑟𝐵−𝐴)(𝑐+𝑑2𝑏−2𝑐−𝑑1−𝐵1−𝐴))₀+𝑐(1−𝐵)1+𝑟₀+(𝑑(1−𝐵)−𝑐(1−𝐴))1−𝑟₀−(1−𝐴)𝑑1−𝑟₀²×1−𝑟₀⁻³=𝐴−𝐵. (28)
The function 𝑓₀ given by (13) shows that the result is sharp. Indeed, at the point 𝑧=𝑟₀ where 𝑟₀ is the root in (0,1) of (26), the function 𝑓₀ satisfies 𝑧𝑓^′₀(𝑧)𝑓₀(𝑧)=()𝑟−1−2𝑐−𝑏₀+2𝑑𝑟²₀1−𝑟₀+𝑑𝑟³₀1−𝑟₀²+2𝑐𝑟₀1−𝑟₀³×𝑐+1+2(𝑐−𝑏)𝑟₀−𝑐1−𝑟₀²−𝑑𝑟²₀1−𝑟₀⁻¹,𝐴−𝐵𝑧𝑓^′₀(𝑧)𝑓₀(𝑧)=(𝑐+1)(𝐴−𝐵)+2(𝑐−𝑏)𝑟₀(𝐴−2𝐵)𝑐+1+2(𝑐−𝑏)𝑟₀−𝑐/1−𝑟₀²−𝑑𝑟²₀/1−𝑟₀−𝑐(𝐴−𝐵)1−𝑟₀²+2𝑐𝑟₀𝐵1−𝑟₀³−𝑑𝑟²₀(𝐴−3𝐵)1−𝑟₀+𝑑𝑟³₀𝐵1−𝑟₀²×𝑐+1+2(𝑐−𝑏)𝑟₀−𝑐1−𝑟₀²−𝑑𝑟²₀1−𝑟₀⁻¹. (29) Then, (26) yields 𝑧𝑓^′₀(𝑧)𝑓₀(𝑧)−1=𝐴−𝐵𝑧𝑓^′₀(𝑧)𝑓₀(𝑧)−1≤𝐵<𝐴≤1,𝑧=𝑟₀, (30) or equivalently 𝑓₀∈𝒮𝒯[𝐴,𝐵].

Theorem 7. Let −1≤𝐵<𝐴≤1. The 𝒮𝒯[𝐴,𝐵]-radius for 𝑓(𝑧)=𝑧+∑^∞_𝑛=2𝑎_𝑛𝑧^𝑛∈𝒜_𝑏 satisfying the coefficient inequality |𝑎_𝑛|≤𝑐/𝑛, 𝑛≥3 and 𝑐>0, is the real root in (0,1) of the equation ()−(2()−(𝑟𝑐𝑐+1)(𝐴−𝐵1−𝐵1−𝐴))2𝑏−2×()()()()1−𝑟=𝑐1−𝐵+𝑐1−𝐴)(1−𝑟log1−𝑟𝑟. (31) This radius is sharp.

Proof. By Lemma 5, condition (27) assures that 𝑟₀ is the 𝒮𝒯[𝐴,𝐵]-radius of 𝑓∈𝒜_𝑏 where 𝑟₀ is the real root of (31). Therefore, using (8) and (19) for 𝑓∈𝒜_𝑏 yields ((1−𝐵)^∞_𝑛=2𝑛−(1−𝐴))𝑎_𝑛𝑟₀^𝑛−1≤2(2(1−𝐵)−(1−𝐴))𝑏𝑟₀+^∞_𝑛=3)(𝑐((1−𝐵𝑛−1−𝐴))𝑛𝑟₀^𝑛−1=2(2(1−𝐵)−(1−𝐴))𝑏𝑟₀()1+𝑐1−𝐵1−𝑟₀−1−𝑟₀−𝑐(1−𝐴)−log1−𝑟₀𝑟₀𝑟−1−⁰2=𝑐(𝐵−𝐴)+(2(1−𝐵)−(1−𝐴))𝑟₀𝑐2𝑏−2+𝑐(1−𝐵)𝑟₀+𝑐(1−𝐴)1−𝑟₀log1−𝑟₀1−𝑟₀𝑟₀=𝐴−𝐵. (32) The result is sharp for the function 𝑓₀ given by (21). Indeed, 𝑓₀ satisfies 𝑧𝑓^′₀(𝑧)𝑓₀(𝑧)=−()𝑟−1𝑐/2−2𝑏₀+𝑐/1−𝑟₀+𝑐log1−𝑟₀/𝑟₀(1+𝑐)+(𝑐/2−2𝑏)𝑟₀+𝑐log1−𝑟₀/𝑟₀,𝐴−𝐵𝑧𝑓^′₀(𝑧)𝑓₀(𝑧)=()+(𝑐1+𝑐)(𝐴−𝐵𝐴−2𝐵)2−2𝑏𝑟₀+𝑐𝐵1−𝑟₀+𝑐𝐴log1−𝑟₀𝑟₀×()+𝑐1+𝑐2−2𝑏𝑟₀+𝑐log1−𝑟₀𝑟₀⁻¹, (33) at the root 𝑧=𝑟₀ in (0,1) of (31). Evidently, the function 𝑓₀ satisfies (30), and hence the result is sharp.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work presented here was supported in parts by an FRGS Grant 203/PMATHS/6711366 and a grant from the University of Delhi.