# Radius Constants for Analytic Functions with Fixed Second Coefficient

Authors: Mahnaz M. Nargesi; Rosihan M. Ali; V. Ravichandran
Journal: The Scientific World Journal
 (2014)

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## Abstract

Letf(z)=z+∑n=2∞anzn 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 an≤cn+d(c,d≥0) or an≤c/n(c>0andn≥3). Other radius constants are also obtained for these functions, and connections with earlier results are made.


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## Body

## 1. Introduction

LetA 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=2∞‍anzn∈S, de Branges [1] obtained the sharp coefficient bound that |an|≤n(n≥2). However, the inequality |an|≤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 ofS 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 |an|≤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 |an|≤1(n≥2) [5–7]. The nth coefficient of close-to-convex functions with argument β satisfies |an|≤1+(n-1)cos⁡β [8], and the coefficients of uniformly starlike functions are bounded by 2/n [9], while |an|≤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 thatf∈A has the Taylor expansion of the form f(z)=z+∑n=2∞‍anzn. Gavrilov [11] showed that the radius of univalence for functions f∈A satisfying |an|≤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 |an|≤M(n≥2) is 1-M/(1+M). The condition |an|≤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 |an|≤n or |an|≤M. Additionally, Yamashita [13] determined that the radius of convexity for functions f∈A satisfying |an|≤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 |an|≤M is the real root of
(1)(M+1)(1-r)3-M(1+r)=0.
Recently, Kalaj et al. [14] obtained the radii of univalence, starlikeness, and convexity for harmonic mappings satisfying certain coefficient inequalities.For two analytic functionsf and g, the function f issubordinate 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).Forβ∈R∖{1},  α≥0, the class L(α,β) consists of functions f∈A satisfying
(2)αz2f′′(z)f(z)+zf′(z)f(z)≺1+(1-2β)z1-z.
Denote by L0(α,β) its subclass consisting of functions f∈A satisfying
(3)|αz2f′′(z)f(z)+zf′(z)f(z)-1|≤|1-β|(β∈R∖{1},α≥0).
These classes were investigated in [15–24].Forβ<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 classST[A,B] of Janowski starlike functions [29] consists of f∈A satisfying the subordination
(4)zf′(z)f(z)≺1+Az1+Bz(-1≤B<A≤1).
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 STβ the class STβ≔L0(0,β)=ST[1-β,0]. Janowski [29] obtained the sharp radius of convexity for ST[A,B].This paper studies the classAb consisting of functions f(z)=z+∑n=2∞‍anzn,  (|a2|=2b,0≤b≤1), in the disk D. The subclass of univalent functions in Ab have been studied in [30–33]. In [33], Ravichandran obtained sharp radii of starlikeness and convexity of order α for functions f∈Ab satisfying |an|≤n or |an|≤M, n≥3. The author also obtained the radius of uniform convexity and parabolic starlikeness for functions f∈Ab satisfying |an|≤n, n≥3.This paper finds radius constants for functionsf(z)=z+∑n=2∞‍anzn∈Ab satisfying either |an|≤cn+d(c,d≥0) or |an|≤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 functionsf∈A to belong to the class L(α,β) is given in the following lemma.Lemma 1 (see [24, 34]).
Letβ∈R∖{1} and α≥0. If f(z)=z+∑n=2∞‍anzn∈A satisfies the inequality
(5)∑n=2∞‍(αn2+(1-α)n-β)|an|≤|1-β|,
then f∈L(α,β).Making use of this lemma, the sharpL(α,β)-radius is obtained for f∈Ab satisfying the coefficient inequality |an|≤cn+d.Theorem 2.
Letβ∈R∖{1}, 6α+3-β≥0, and α≥0. The L(α,β)-radius for f(z)=z+∑n=2∞‍anzn∈Ab satisfying the coefficient inequality |an|≤cn+d, c,d≥0, n≥3, is the real root in (0,1) of the equation
(6)((c+d)(1-β)+|1-β|+(2α+2-β)(2(c-b)+d)r)(1-r)4=cα(1+4r+r2)+((1-α)c+αd)(1-r2)+((1-α)d-βc)(1-r)2-βd(1-r)3.
For β<1, this number is also the L0(α,β)-radius of f∈Ab. The results are sharp.Proof.
The numberr0 is the L(α,β)-radius for f∈Ab if and only if f(r0z)/r0∈L(α,β). Therefore, by Lemma 1, it is sufficient to verify the inequality
(7)∑n=2∞‍(αn2+(1-α)n-β)|an|r0n-1≤|1-β|,
where r0 is the real root in (0,1) of (6). Using the known expansions
(8)∑n=3∞‍r0n-1=11-r0-1-r0,(9)∑n=3∞‍nr0n-1=1(1-r0)2-1-2r0,(10)∑n=3∞‍n2r0n-1=1+r0(1-r0)3-1-4r0,(11)∑n=3∞‍n3r0n-1=1+4r0+r02(1-r0)4-1-8r0
leads to
(12)∑n=2∞‍(αn2+(1-α)n-β)|an|r0n-1≤2(2α+2-β)br0+∑n=3∞‍(αn2+(1-α)n-β)(cn+d)r0n-1=2(2α+2-β)br0+cα(1+4r0+r02(1-r0)4-1-8r0)+((1-α)c+αd)(1+r0(1-r0)3-1-4r0)+((1-α)d-βc)(1(1-r0)2-1-2r0)-βd(11-r0-1-r0)=(c+d)(β-1)-(2α+2-β)(2(c-b)+d)r0+(cα(1+4r0+r02)+((1-α)c+αd)(1-r02)+((1-α)d-βc)(1-r0)2-βd(1-r0)3)×(1-r0)-4.=|1-β|.
Forβ<1, consider the function
(13)f0(z)=z-2bz2-∑n=3∞‍(cn+d)zn=(c+1)z+2(c-b)z2-cz(1-z)2-dz31-z.
At the root z=r0 in (0,1) of (6), f0 satisfies
(14)Re(αz2f′′0(z)f0(z)+zf0′(z)f0(z))=1-N(r0)D(r0)=β,
where
(15)N(r0)=-2(c-b)(2α+1)r0+2cr0(2α+1)(1-r0)3+6cαr02(1-r0)4+2dr02(3α+1)1-r0+dr03(6α+1)(1-r0)2+2dr04α(1-r0)3,D(r0)=c+1+2(c-b)r0-c(1-r0)2-dr021-r0.
This shows that r0 is the sharp L(α,β)-radius for f∈Ab. For β<1, (14) shows that the rational expression N(r0)/D(r0) is positive, and therefore the equality
(16)|αz2f0′′(z)f0(z)+zf0′(z)f0(z)-1|=1-β
holds. Thus, r0 is the sharp L0(α,β)-radius for f∈Ab when β<1.
Forβ>1, the function
(17)f0(z)=z+2bz2+∑n=3∞‍(cn+d)zn=(1-c)z+2(b-c)z2+cz(1-z)2+dz31-z
demonstrates sharpness of the result. The derivation is similar to the case β<1 and is omitted.Theorem 3.
Letβ∈R∖{1} and α≥0. The L(α,β)-radius of f(z)=z+∑n=2∞‍anzn∈Ab satisfying the coefficient inequality |an|≤c/n for n≥3 and c>0 is the real root in (0,1) of the equation
(18)[c(1-β)+|1-β|+(2α+2-β)r(c2-2b)](1-r)2=cα+(1-α)c(1-r)+βc(1-r)2log⁡(1-r)r.
For β<1, this number is also the L0(α,β)-radius of f∈Ab. The results are sharp.Proof.
By Lemma1, r0 is the L(α,β)-radius of functions f∈Ab when inequality (7) holds for the real root r0 of (18) in (0,1). Using (8) and (9) together with
(19)∑n=3∞‍r0n-1n=-log⁡(1-r0)r0-1-r02
leads to
(20)∑n=2∞‍(αn2+(1-α)n-β)|an|r0n-1≤2(2α+2-β)br0+∑n=3∞‍(αn2+(1-α)n-β)(cn)r0n-1=2(2α+2-β)br0+cα(1(1-r0)2-1-2r0)+(1-α)c(11-r0-1-r0)-βc(-log⁡⁡(1-r0)r0-1-r02)=c(β-1)+(2α+2-β)r0(2b-c2)+cαr0+(1-α)c(1-r0)r0+βc(1-r0)2log⁡⁡(1-r0)(1-r0)2r0=|1-β|.
To verify sharpness forβ<1, consider the function
(21)f0(z)=z-2bz2-∑n=3∞‍cnzn=(1+c)z+(c2-2b)z2+clog⁡(1-z).
At the root z=r0 in (0,1) of (18), f0 satisfies
(22)Re(αz2f0′′(z)f0(z)+zf0′(z)f0(z))=1-(clog⁡(1-r0)r0-(c2-2b)r0(2α+1)+cr0α(1-r0)2+c1-r0+clog⁡(1-r0)r0-(c2-2b)r0(2α+1)+cr0α(1-r0)2)×((1+c)+(c2-2b)r0+clog⁡(1-r0)r0)-1=β.
Thus, r0 is the sharp L(α,β)-radius for f∈Ab. For β<1, the rational expression in (22) is positive, and therefore
(23)|αz2f0′′(z)f0(z)+zf0′(z)f0(z)-1|=1-β,
which shows that r0 is the sharp L0(α,β)-radius for f∈Ab. For β>1, sharpness of the result is demonstrated by the function f0 given by
(24)f0(z)=z+2bz2+∑n=3∞‍cnzn=(1-c)z+(2b-c2)z2-clog⁡(1-z).Remark 4.
The results obtained above yield the following special cases.(1)
Forα=0, β=0, c=1, d=0, and 0≤b≤1, Theorem 2 yields the radius of starlikeness obtained by Yamashita [13].(2)
Forα=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)
Forα=0, Theorem 3 yields the radius of starlikeness of order β for f∈Ab 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=2∞‍anzn∈A satisfies the inequality
(25)∑n=2∞‍((1-B)n-(1-A))|an|≤A-B,
then f∈ST[A,B].The next result finds the sharpST[A,B]-radius for f∈Ab satisfying the coefficient inequality |an|≤cn+d.Theorem 6.
Let-1≤B<A≤1. The ST[A,B]-radius for f(z)=z+∑n=2∞‍anzn∈Ab satisfying the coefficient inequality |an|≤cn+d, n≥3 and c,d≥0, is the real root in (0,1) of the equation
(26)[(A-B)(c+d+1)-(2b-2c-d)(2(1-B)-(1-A))r](1-r)3=c(1-B)(1+r)+(d(1-B)-c(1-A))(1-r)-(1-A)d(1-r)2.
This radius is sharp.Proof.
It is evident thatr0 is the ST[A,B]-radius of f∈Ab if and only if f(r0z)/r0∈ST[A,B]. Hence, by Lemma 5, it suffices to show that
(27)∑n=2∞‍((1-B)n-(1-A))|an|r0n-1≤A-B(-1≤B<A≤1),
where r0 is the root in (0,1) of (26). From (8), (9), and (10), it follows that
(28)∑n=2∞‍((1-B)n-(1-A))|an|r0n-1≤2(2(1-B)-(1-A))br0+∑n=3∞‍((1-B)n-(1-A))(cn+d)r0n-1=2(2(1-B)-(1-A))br0+c(1-B)(1+r0(1-r0)3-1-4r0)+(d(1-B)-c(1-A))(1(1-r0)2-1-2r0)-(1-A)d(11-r0-1-r0)=(B-A)(c+d)+(2b-2c-d)×(2(1-B)-(1-A))r0+((1-r0)2c(1-B)(1+r0)+(d(1-B)-c(1-A))(1-r0)-(1-A)d(1-r0)2)×(1-r0)-3=A-B.
The functionf0 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
(29)|zf0′(z)f0(z)-1|=(-2(c-b)r0+2dr021-r0+dr03(1-r0)2+2cr0(1-r0)3)×(c+1+2(c-b)r0-c(1-r0)2-dr021-r0)-1,|A-Bzf0′(z)f0(z)|=(c+1)(A-B)+2(c-b)r0(A-2B)c+1+2(c-b)r0-c/(1-r0)2-dr02/(1-r0)-((dr03B/(1-r0)2)c(A-B)(1-r0)2+2cr0B(1-r0)3-dr02(A-3B)1-r0+dr03B(1-r0)2)×(c+1+2(c-b)r0-c(1-r0)2-dr021-r0)-1.
Then, (26) yields
(30)|zf0′(z)f0(z)-1|=|A-Bzf0′(z)f0(z)|(-1≤B<A≤1,z=r0),
or equivalently f0∈ST[A,B].Theorem 7.
Let-1≤B<A≤1. The ST[A,B]-radius for f(z)=z+∑n=2∞‍anzn∈Ab satisfying the coefficient inequality |an|≤c/n, n≥3 and c>0, is the real root in (0,1) of the equation
(31)((c+1)(A-B)-(2(1-B)-(1-A))r(2b-c2))×(1-r)=c(1-B)+c(1-A)(1-r)log⁡⁡(1-r)r.
This radius is sharp.Proof.
By Lemma5, condition (27) assures that r0 is the ST[A,B]-radius of f∈Ab where r0 is the real root of (31). Therefore, using (8) and (19) for f∈Ab yields
(32)∑n=2∞‍((1-B)n-(1-A))|an|r0n-1≤2(2(1-B)-(1-A))br0+∑n=3∞‍((1-B)n-(1-A))(cn)r0n-1=2(2(1-B)-(1-A))br0+c(1-B)(11-r0-1-r0)-c(1-A)(-log⁡⁡(1-r0)r0-1-r02)=c(B-A)+(2(1-B)-(1-A))r0(2b-c2)+c(1-B)r0+c(1-A)(1-r0)log⁡⁡(1-r0)(1-r0)r0=A-B.
The result is sharp for the function f0 given by (21). Indeed, f0 satisfies
(33)|zf0′(z)f0(z)-1|=-(c/2-2b)r0+c/(1-r0)+(clog⁡(1-r0))/r0(1+c)+(c/2-2b)r0+(clog⁡(1-r0))/r0,|A-Bzf0′(z)f0(z)|=(cAlog⁡(1-r0)r0(1+c)(A-B)+(A-2B)(c2-2b)r0+cB1-r0+cAlog⁡(1-r0)r0)×((1+c)+(c2-2b)r0+clog⁡(1-r0)r0)-1,
at the root z=r0 in (0,1) of (31). Evidently, the function f0 satisfies (30), and hence the result is sharp.



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