# Radius Constants for Functions with the Prescribed Coefficient Bounds

Authors: Om P. Ahuja; Sumit Nagpal; V. Ravichandran
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

For an analytic univalent functionf(z)=z+∑n=2∞anzn in the unit disk, it is well-known that an≤n for n≥2. But the inequality an≤n does not imply the univalence of f. This motivated several authors to determine various radii constants associated with the analytic functions having prescribed coefficient bounds. In this paper, a survey of the related work is presented for analytic and harmonic mappings. In addition, we establish a coefficient inequality for sense-preserving harmonic functions to compute the bounds for the radius of univalence, radius of full starlikeness/convexity of order α  (0≤α<1) for functions with prescribed coefficient bound on the analytic part.


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## Body

## 1. Introduction

LetA denote the class of all analytic functions f defined in the open unit disk D:={z∈C:|z|<1} normalized by f(0)=0=f′(0)-1. For functions f of the form
(1)f(z)=z+∑n=2∞anzn
belonging to the subclass S of A consisting of univalent functions, de Branges [1] proved the famous Bieberbach conjecture that |an|≤n for n≥2. However, the inequality |an|≤n (n≥2) does not imply that f is univalent. A function f given by (1) whose coefficients satisfy |an|≤n for n≥2 is necessarily analytic in D by the usual comparison test and hence a member of A. But it need not be univalent. For example, the function
(2)f(z)=z-2z2-3z3-⋯=2z-z(1-z)2
satisfies the inequality |an|≤n (n≥2) but its derivative vanishes inside D and so the function f is not univalent in D. It is therefore of interest to determine the largest subdisk |z|<ρ<1 in which the functions f satisfying the inequality |an|≤n are univalent. Motivated by this problem, various radii problems associated with analytic as well as harmonic functions having prescribed coefficient bounds have been studied and we present a brief review of the research on this topic. Recall that given two subsets F and G of A, the G-radius in F is the largest R such that, for every f∈F, r-1f(rz)∈G for each r≤R.


### 1.1. Analytic Case

Most of the classes in univalent function theory are characterized by the quantitieszf′(z)/f(z) or 1+zf′′(z)/f′(z) lying in a given domain in the right half-plane. For instance, the subclasses S*(α) and K(α)   (0≤α<1) of S consisting of starlike functions of order α and convex functions of order α, respectively, are defined analytically by the equivalences
(3)f∈S*(α)⟺Re(zf′(z)f(z))>α,f∈K(α)⟺Re(zf′′(z)f′(z)+1)>α.
These classes were introduced by Robertson [2]. The classes S*:=S*(0) and K:=K(0) are the familiar classes of starlike and convex functions, respectively. Goodman [3] introduced the class UCV of uniformly convex functions f∈A, which map every circular arc γ contained in D with center ζ∈D onto a convex arc. For f∈A, Rønning [4] and Ma and Minda [5] independently proved that
(4)f∈UCV⟺Re(zf′′(z)f′(z)+1)>|zf′′(z)f′(z)|(z∈D).
Closely related to the class UCV is the class SP of parabolic starlike functions, introduced by Rønning [4] consisting of functions f=zg′ where g∈UCV; that is, a function f∈SP satisfies
(5)Re(zf′(z)f(z))>|zf′(z)f(z)-1|(z∈D).In 1970, Gavrilov [6] 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. In 1982, Yamashita [7] showed that the radius of univalence obtained by Gavrilov [6] is also the radius of starlikeness for functions f∈A satisfying |an|≤n. Yamashita [7] also proved that the radius of convexity for functions f∈A satisfying |an|≤n   (n≥2) is the real root r0≃0.090 of the equation 2(1-r)4-(1+4r+r2)=0.The inequality|an|≤M holds for functions f∈A satisfying |f(z)|≤M. Gavrilov [6] proved that the radius of univalence for functions f∈A satisfying |an|≤M   (n≥2) is 1-M/(1+M), which also turned out to be their radius of starlikeness, a result proved by Yamashita [7]. The radius of convexity for functions f∈A satisfying |an|≤M   (n≥2) is the real root of the equation (M+1)(1-r)3-M(1+r)=0.For0≤b≤1, let Ab denote the class of functions f given by (1) with |a2|=2b. Since the second coefficient of normalized univalent functions determines their important properties such as Koebe-one-quarter theorem, growth and distortion theorems, the last author [8] obtained the sharp S*(α), K(α)   (0≤α<1), UCV and SP radii for functions f∈Ab satisfying |an|≤n, |an|≤M, or |an|≤M/n   (M>0) for n≥3. Observe that a function f∈A with Ref′(z)>0 satisfies |an|≤2/n for n≥2. Indeed, Ravichandran [8] proved the following theorem, which includes the results of Gavrilov [6] and Yamashita [7] as special cases.Theorem 1 (see [8]).
Letf∈Ab be given by (1) with |an|≤n for n≥3. Then we have the following.(i)
f satisfies the inequality
(6)|zf′(z)f(z)-1|<1-α
in |z|<r0 where r0=r0(α) is the real root in (0,1) of the equation 1-α+(1+α)r=2(1-α+(2-α)(1-b)r)(1-r)3. In particular, the number r0(α) is also the radius of starlikeness of order α and the number r0(1/2) is the radius of parabolic starlikeness of the given functions.(ii)
f satisfies the inequality
(7)|zf′′(z)f′(z)|<1-α
in |z|<s0 where s0=s0(α) is the real root in (0,1) of the equation 2(1-α+2(2-α)(1-b)r)(1-r)4=1-α+4r+(1+α)r2. In particular, the number s0(α) is also the radius of convexity of order α and the number s0(1/2) is the radius of uniform convexity of the given functions.The results are sharp for the function
(8)f0(z)=2z+2(1-b)z2-z(1-z)2f0(z)=z-2bz2-3z3-4z4-⋯.It is observed that [9] if a function f∈A satisfies Re(f′(z)+zf′′(z))>0 for z∈D, then |an|≤2/n2. Similarly, Reade [10] proved that a close-to-star function f∈A satisfies |an|≤n2 for n≥2. However, the converse in both the cases is not true, in general. Recently, Mendiratta et al. [11] obtained sharp radii of starlikeness of order α (0≤α<1), convexity of order α   (0≤α<1), parabolic starlikeness and uniform convexity for the class Ab when |an|≤M/n2 or |an|≤Mn2   (M>0) for n≥3. Ali et al. [12] also worked in the similar direction and obtained similar radii constants.


### 1.2.  Harmonic Case

In a simply connected domainΩ⊂C, a complex-valued harmonic function f has the representation f=h+g¯, where h and g are analytic in Ω. We call the functions h and g the analytic and the coanalytic parts of f, respectively. Let H denote the class of all harmonic functions f=h+g¯ in D normalized so that h and g take the form
(9)h(z)=z+∑n=2∞anzn,g(z)=∑n=1∞bnzn.
Since the Jacobian of f is given by Jf=|h′|2-|g′|2, by a theorem of Lewy [13], f is sense-preserving if and only if |g′|<|h′|, or equivalently if h′(z)≠0 and the second dilatation wf=g′/h′ satisfies |wf(z)|<1 in D. Let Hsp be the subclass of H consisting of sense-preserving functions. Then it is easy to see that |b1|<1 for functions in the class Hsp. Set H0:={f∈H:b1=0} and Hsp0:=Hsp∩H0. Finally, let SH and SH0 be subclasses of Hsp and Hsp0, respectively, consisting of univalent functions.One of the important questions in the study of classSH0 and its subclasses is related to coefficient bounds. In 1984, Clunie and Sheil-Small [14] conjectured that the Taylor coefficients of the series of h and g satisfy the inequality
(10)|an|≤16(2n+1)(n+1),|bn|≤16(2n-1)(n-1),∀n≥2
and it is still open. These researchers proposed this coefficient conjecture because the harmonic Koebe function K=H+G¯ where
(11)H(z)=z-(1/2)z2+(1/6)z3(1-z)3=z+16∑n=2∞(n+1)(2n+1)zn,G(z)=(1/2)z2+(1/6)z3(1-z)3=16∑n=2∞(n-1)(2n-1)zn
is expected to play the extremal role in the class SH0. However, this conjecture is proved for all functions f∈SH0 with real coefficients and all functions f∈SH0 for which either f(D) is starlike with respect to the origin, close-to-convex, or convex in one direction (see [14–16]).Iff∈SH0 for which f(D) is convex, Clunie and Sheil-Small [14] proved that the Taylor coefficients of h and g satisfy the inequalities
(12)|an|≤n+12,|bn|≤n-12,∀n≥2,
and equality occurs for the harmonic half-plane mapping
(13)L(z)=M(z)+N(z)¯,M(z):=z-(1/2)z2(1-z)2,N(z):=-(1/2)z2(1-z)2.LetKH0 and SH*0 be subclasses of SH0 consisting of functions f for which f(D) is convex and f(D) is starlike with respect to origin, respectively. Recall that convexity and starlikeness are not hereditary properties for univalent harmonic mappings (see [17–19]). Chuaqui et al. [19] introduced the notion of fully starlike and fully convex harmonic functions that do inherit the properties of starlikeness and convexity, respectively. The last two authors [18] generalized this concept to fully starlike functions of order α and fully convex harmonic functions of order α for 0≤α<1. Let FSH*(α) and FKH(α)   (0≤α<1) be subclasses of SH consisting of fully starlike functions of order α and fully convex functions of order α, with FSH*:=FSH*(0) and FKH:=FKH(0). The functions in the classes FSH*(α) and FKH(α) are characterized by the conditions (∂/∂θ)arg⁡f(reiθ)>α and (∂/∂θ)(arg⁡{(∂/∂θ)f(reiθ)})>α for every circle |z|=r, z=reiθ, respectively, where 0≤θ<2π, 0<r<1.The radius of full convexity of the classKH0 is 2-1 while the radius of full convexity of the class SH*0 is 3-8 (see [14, 16, 20]). The corresponding problems for the radius of full starlikeness are still unsolved. However, Kalaj et al. [21] worked in this direction and determined the radius of univalence and full starlikeness of functions f=h+g¯ whose coefficients satisfy the conditions (10) and (12). This, in turn, provides a bound for the radius of full starlikeness for convex and starlike mappings in SH0. These results are generalized in context of fully starlike and fully convex functions of order α (0≤α<1) in [18]. The authors [18] proved the following result.Theorem 2 (see [18]).
Leth and g have the form (9) with b1=g′(0)=0 and 0≤α<1. Then we have the following.(a)
If the coefficients of the series satisfy the conditions (10), then f=h+g¯ is univalent and fully starlike of order α in the disk |z|<rS, where rS=rS(α) is the real root in (0,1) of the equation 2(1-α)(1-r)4+α(1-r)2-(1+r)2=0.(b)
If the coefficients of the series satisfy the conditions (12), then f=h+g¯ is univalent and fully starlike of order α in the disk |z|<rS, where rS=rS(α) is the real root in (0,1) of the equation (2-α)(1-r)3+αr(1-r)2-1-r=0.Moreover, the results are sharp for each α∈[0,1).Theorem2 gives the bounds for the radius of full starlikeness of order α (0≤α<1) for the classes SH*0 and KH0. In addition, the authors in [18] also determined the bounds for the radius of full convexity of order α   (0<α<1) for these classes.The analytic part of harmonic mappings plays a vital role in shaping their geometric properties. For instance, iff=h+g¯∈Hsp and h is convex univalent, then f∈SH and maps D onto a close-to-convex domain (see [14, Theorem 5.17, p. 20]). However, if f=h+g¯∈Hsp where h and g are given by (9) and |an|≤1 for n≥2, then f need not be even univalent; for example, the function
(14)f(z)=z-z22+z22-z33¯,z∈D
belongs to Hsp but is not univalent in D since f(z0)=f(z¯0)=3/4 where z0=(3+3i)/4∈D. Note that a convex univalent function z+a2z2+a3z3+⋯ satisfies |an|≤1 for n=2,3,…. This paper aims to determine the coefficient inequalities and radius constants for certain subfamilies of Hsp with the prescribed coefficient bound on the analytic part.A coefficient inequality for functions in the classHsp is obtained in Section 2 which, in particular, improves the coefficient inequality proved by Polatoğlu et al. [22] for perturbed harmonic mappings. Using this inequality, the bounds for the radius of univalence, full starlikeness, and full convexity of order α (0≤α<1) are obtained for functions f=h+g¯∈Hsp0 where the coefficients of the analytic part h satisfy one of the conditions |an|≤n, |an|≤1, or |an|≤1/n for n≥2. In addition, we will also discuss a case under which these bounds can be improved.In the third section, sharp bounds onβ (depending upon α and |b1|) are determined for a function f=h+g¯∈H, where h and g are given by (9), satisfying either of the following two conditions:
(15)∑n=2∞n|an|+∑n=1∞n|bn|≤βor∑n=2∞n2|an|+∑n=1∞n2|bn|≤β,
to be either fully starlike of order α or fully convex of order α. The obtained results are applied to hypergeometric functions in Section 4.


## 1.1. Analytic Case

Most of the classes in univalent function theory are characterized by the quantitieszf′(z)/f(z) or 1+zf′′(z)/f′(z) lying in a given domain in the right half-plane. For instance, the subclasses S*(α) and K(α)   (0≤α<1) of S consisting of starlike functions of order α and convex functions of order α, respectively, are defined analytically by the equivalences
(3)f∈S*(α)⟺Re(zf′(z)f(z))>α,f∈K(α)⟺Re(zf′′(z)f′(z)+1)>α.
These classes were introduced by Robertson [2]. The classes S*:=S*(0) and K:=K(0) are the familiar classes of starlike and convex functions, respectively. Goodman [3] introduced the class UCV of uniformly convex functions f∈A, which map every circular arc γ contained in D with center ζ∈D onto a convex arc. For f∈A, Rønning [4] and Ma and Minda [5] independently proved that
(4)f∈UCV⟺Re(zf′′(z)f′(z)+1)>|zf′′(z)f′(z)|(z∈D).
Closely related to the class UCV is the class SP of parabolic starlike functions, introduced by Rønning [4] consisting of functions f=zg′ where g∈UCV; that is, a function f∈SP satisfies
(5)Re(zf′(z)f(z))>|zf′(z)f(z)-1|(z∈D).In 1970, Gavrilov [6] 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. In 1982, Yamashita [7] showed that the radius of univalence obtained by Gavrilov [6] is also the radius of starlikeness for functions f∈A satisfying |an|≤n. Yamashita [7] also proved that the radius of convexity for functions f∈A satisfying |an|≤n   (n≥2) is the real root r0≃0.090 of the equation 2(1-r)4-(1+4r+r2)=0.The inequality|an|≤M holds for functions f∈A satisfying |f(z)|≤M. Gavrilov [6] proved that the radius of univalence for functions f∈A satisfying |an|≤M   (n≥2) is 1-M/(1+M), which also turned out to be their radius of starlikeness, a result proved by Yamashita [7]. The radius of convexity for functions f∈A satisfying |an|≤M   (n≥2) is the real root of the equation (M+1)(1-r)3-M(1+r)=0.For0≤b≤1, let Ab denote the class of functions f given by (1) with |a2|=2b. Since the second coefficient of normalized univalent functions determines their important properties such as Koebe-one-quarter theorem, growth and distortion theorems, the last author [8] obtained the sharp S*(α), K(α)   (0≤α<1), UCV and SP radii for functions f∈Ab satisfying |an|≤n, |an|≤M, or |an|≤M/n   (M>0) for n≥3. Observe that a function f∈A with Ref′(z)>0 satisfies |an|≤2/n for n≥2. Indeed, Ravichandran [8] proved the following theorem, which includes the results of Gavrilov [6] and Yamashita [7] as special cases.Theorem 1 (see [8]).
Letf∈Ab be given by (1) with |an|≤n for n≥3. Then we have the following.(i)
f satisfies the inequality
(6)|zf′(z)f(z)-1|<1-α
in |z|<r0 where r0=r0(α) is the real root in (0,1) of the equation 1-α+(1+α)r=2(1-α+(2-α)(1-b)r)(1-r)3. In particular, the number r0(α) is also the radius of starlikeness of order α and the number r0(1/2) is the radius of parabolic starlikeness of the given functions.(ii)
f satisfies the inequality
(7)|zf′′(z)f′(z)|<1-α
in |z|<s0 where s0=s0(α) is the real root in (0,1) of the equation 2(1-α+2(2-α)(1-b)r)(1-r)4=1-α+4r+(1+α)r2. In particular, the number s0(α) is also the radius of convexity of order α and the number s0(1/2) is the radius of uniform convexity of the given functions.The results are sharp for the function
(8)f0(z)=2z+2(1-b)z2-z(1-z)2f0(z)=z-2bz2-3z3-4z4-⋯.It is observed that [9] if a function f∈A satisfies Re(f′(z)+zf′′(z))>0 for z∈D, then |an|≤2/n2. Similarly, Reade [10] proved that a close-to-star function f∈A satisfies |an|≤n2 for n≥2. However, the converse in both the cases is not true, in general. Recently, Mendiratta et al. [11] obtained sharp radii of starlikeness of order α (0≤α<1), convexity of order α   (0≤α<1), parabolic starlikeness and uniform convexity for the class Ab when |an|≤M/n2 or |an|≤Mn2   (M>0) for n≥3. Ali et al. [12] also worked in the similar direction and obtained similar radii constants.


## 1.2.  Harmonic Case

In a simply connected domainΩ⊂C, a complex-valued harmonic function f has the representation f=h+g¯, where h and g are analytic in Ω. We call the functions h and g the analytic and the coanalytic parts of f, respectively. Let H denote the class of all harmonic functions f=h+g¯ in D normalized so that h and g take the form
(9)h(z)=z+∑n=2∞anzn,g(z)=∑n=1∞bnzn.
Since the Jacobian of f is given by Jf=|h′|2-|g′|2, by a theorem of Lewy [13], f is sense-preserving if and only if |g′|<|h′|, or equivalently if h′(z)≠0 and the second dilatation wf=g′/h′ satisfies |wf(z)|<1 in D. Let Hsp be the subclass of H consisting of sense-preserving functions. Then it is easy to see that |b1|<1 for functions in the class Hsp. Set H0:={f∈H:b1=0} and Hsp0:=Hsp∩H0. Finally, let SH and SH0 be subclasses of Hsp and Hsp0, respectively, consisting of univalent functions.One of the important questions in the study of classSH0 and its subclasses is related to coefficient bounds. In 1984, Clunie and Sheil-Small [14] conjectured that the Taylor coefficients of the series of h and g satisfy the inequality
(10)|an|≤16(2n+1)(n+1),|bn|≤16(2n-1)(n-1),∀n≥2
and it is still open. These researchers proposed this coefficient conjecture because the harmonic Koebe function K=H+G¯ where
(11)H(z)=z-(1/2)z2+(1/6)z3(1-z)3=z+16∑n=2∞(n+1)(2n+1)zn,G(z)=(1/2)z2+(1/6)z3(1-z)3=16∑n=2∞(n-1)(2n-1)zn
is expected to play the extremal role in the class SH0. However, this conjecture is proved for all functions f∈SH0 with real coefficients and all functions f∈SH0 for which either f(D) is starlike with respect to the origin, close-to-convex, or convex in one direction (see [14–16]).Iff∈SH0 for which f(D) is convex, Clunie and Sheil-Small [14] proved that the Taylor coefficients of h and g satisfy the inequalities
(12)|an|≤n+12,|bn|≤n-12,∀n≥2,
and equality occurs for the harmonic half-plane mapping
(13)L(z)=M(z)+N(z)¯,M(z):=z-(1/2)z2(1-z)2,N(z):=-(1/2)z2(1-z)2.LetKH0 and SH*0 be subclasses of SH0 consisting of functions f for which f(D) is convex and f(D) is starlike with respect to origin, respectively. Recall that convexity and starlikeness are not hereditary properties for univalent harmonic mappings (see [17–19]). Chuaqui et al. [19] introduced the notion of fully starlike and fully convex harmonic functions that do inherit the properties of starlikeness and convexity, respectively. The last two authors [18] generalized this concept to fully starlike functions of order α and fully convex harmonic functions of order α for 0≤α<1. Let FSH*(α) and FKH(α)   (0≤α<1) be subclasses of SH consisting of fully starlike functions of order α and fully convex functions of order α, with FSH*:=FSH*(0) and FKH:=FKH(0). The functions in the classes FSH*(α) and FKH(α) are characterized by the conditions (∂/∂θ)arg⁡f(reiθ)>α and (∂/∂θ)(arg⁡{(∂/∂θ)f(reiθ)})>α for every circle |z|=r, z=reiθ, respectively, where 0≤θ<2π, 0<r<1.The radius of full convexity of the classKH0 is 2-1 while the radius of full convexity of the class SH*0 is 3-8 (see [14, 16, 20]). The corresponding problems for the radius of full starlikeness are still unsolved. However, Kalaj et al. [21] worked in this direction and determined the radius of univalence and full starlikeness of functions f=h+g¯ whose coefficients satisfy the conditions (10) and (12). This, in turn, provides a bound for the radius of full starlikeness for convex and starlike mappings in SH0. These results are generalized in context of fully starlike and fully convex functions of order α (0≤α<1) in [18]. The authors [18] proved the following result.Theorem 2 (see [18]).
Leth and g have the form (9) with b1=g′(0)=0 and 0≤α<1. Then we have the following.(a)
If the coefficients of the series satisfy the conditions (10), then f=h+g¯ is univalent and fully starlike of order α in the disk |z|<rS, where rS=rS(α) is the real root in (0,1) of the equation 2(1-α)(1-r)4+α(1-r)2-(1+r)2=0.(b)
If the coefficients of the series satisfy the conditions (12), then f=h+g¯ is univalent and fully starlike of order α in the disk |z|<rS, where rS=rS(α) is the real root in (0,1) of the equation (2-α)(1-r)3+αr(1-r)2-1-r=0.Moreover, the results are sharp for each α∈[0,1).Theorem2 gives the bounds for the radius of full starlikeness of order α (0≤α<1) for the classes SH*0 and KH0. In addition, the authors in [18] also determined the bounds for the radius of full convexity of order α   (0<α<1) for these classes.The analytic part of harmonic mappings plays a vital role in shaping their geometric properties. For instance, iff=h+g¯∈Hsp and h is convex univalent, then f∈SH and maps D onto a close-to-convex domain (see [14, Theorem 5.17, p. 20]). However, if f=h+g¯∈Hsp where h and g are given by (9) and |an|≤1 for n≥2, then f need not be even univalent; for example, the function
(14)f(z)=z-z22+z22-z33¯,z∈D
belongs to Hsp but is not univalent in D since f(z0)=f(z¯0)=3/4 where z0=(3+3i)/4∈D. Note that a convex univalent function z+a2z2+a3z3+⋯ satisfies |an|≤1 for n=2,3,…. This paper aims to determine the coefficient inequalities and radius constants for certain subfamilies of Hsp with the prescribed coefficient bound on the analytic part.A coefficient inequality for functions in the classHsp is obtained in Section 2 which, in particular, improves the coefficient inequality proved by Polatoğlu et al. [22] for perturbed harmonic mappings. Using this inequality, the bounds for the radius of univalence, full starlikeness, and full convexity of order α (0≤α<1) are obtained for functions f=h+g¯∈Hsp0 where the coefficients of the analytic part h satisfy one of the conditions |an|≤n, |an|≤1, or |an|≤1/n for n≥2. In addition, we will also discuss a case under which these bounds can be improved.In the third section, sharp bounds onβ (depending upon α and |b1|) are determined for a function f=h+g¯∈H, where h and g are given by (9), satisfying either of the following two conditions:
(15)∑n=2∞n|an|+∑n=1∞n|bn|≤βor∑n=2∞n2|an|+∑n=1∞n2|bn|≤β,
to be either fully starlike of order α or fully convex of order α. The obtained results are applied to hypergeometric functions in Section 4.


## 2. A Coefficient Inequality and Radius Constants

Firstly, we will obtain a coefficient inequality for functions in the classHsp.Theorem 3.
Letf=h+g¯∈Hsp, where h and g are given by (9). Then
(16)|bn|≤|b1||an|+(1-|b1|2)n∑k=1n-1k|ak|,
for n≥2, with a1=1. In particular, one has
(17)|bn|≤1n∑k=1nk|ak|,n=2,3,….Proof.
Sincef∈Hsp, the function w(z)=g′(z)/h′(z)=∑n=0∞‍wnzn is analytic in D and |w(z)|<1 in D. On equating the coefficients of zn-1 in g′(z)=w(z)h′(z), we obtain
(18)nbn=a1wn-1+2wn-2a2+3wn-3a3nbn=+⋯+(n-1)w1an-1+nw0an,
where a1=1. Since |wn|≤1-|w0|2 (see [23, p. 172]), it immediately follows that
(19)n|bn|≤(1-|w0|2)∑k=1n-1k|ak|+n|w0||an|,(a1=1).
Since w0=g′(0)/h′(0)=b1, the desired result follows.For specific choices of the analytic parth in a harmonic function f=h+g¯∈Hsp, Theorem 3 yields the following result.Corollary 4.
Letf=h+g¯∈Hsp, where h and g are given by (9). Then we have the following. (i)
If|an|≤n or, in particular, h is univalent, then |bn|≤(2n+1)(n+1)/6, n=2,3,….(ii)
If|an|≤1 or, in particular, h is convex univalent, then |bn|≤(n+1)/2, n=2,3,….(iii)
If|an|≤1/n or, in particular, Reh′(z)>0, then |bn|≤1, n=2,3,….Remark 5.
Polatoğlu et al. [22] determined the coefficient inequality for harmonic functions in a subclass of Hsp, for which the analytic part is a univalent function in D. They proved that if f=h+g¯∈Hsp where h and g are given by (9) and if h is univalent in D, then
(20)|bn|≤1n(2n6-n2-4n-6),n=1,2,….
It is evident that Corollary 4(i) improves this bound.Now, we will determine the radius of univalence, radius of full starlikeness/full convexity of orderα   (0≤α<1) for the class Hsp0 with specific choices of the coefficient bound on the analytic part. We will make use of the following sufficient coefficient conditions for a harmonic function to be in the classes FSH*(α) and FKH(α) (0≤α<1) that directly follow from the corresponding results in [24, 25].Lemma 6 (see [24, 25]).
Letf=h+g¯, where h and g are given by (9) and let 0≤α<1. Then we have the following.(i)
If(21)∑n=2∞n-α1-α|an|+∑n=1∞n+α1-α|bn|≤1,
then f∈FSH*(α).(ii)
If(22)∑n=2∞n(n-α)1-α|an|+∑n=1∞n(n+α)1-α|bn|≤1,
then f∈FKH(α).Theorem 7.
Letf=h+g¯∈Hsp0, where h and g are given by (9) with b1=g′(0)=0 and 0≤α<1. Then we have the following. (i)
If|an|≤n or, in particular, h is univalent, then f is univalent and fully starlike of order α in the disk |z|<r1 where r1=r1(α) is the real root of the equation
(23)12(1-α)r4+(49α-48)r3+8(9-8α)r2+3(11α-18)r+6(1-α)=0
in the interval (0,1).(ii)
If|an|≤1 or, in particular, h is convex univalent, then f is univalent and fully starlike of order α in the disk |z|<r2 where r2=r2(α) is the real root of the equation
(24)4(1-α)r3+(9α-12)r2+(12-7α)r-2(1-α)=0
in the interval (0,1).(iii)
If|an|≤1/n or, in particular, Reh′(z)>0, then f is univalent and fully starlike of order α in the disk |z|<r3 where r3=r3(α) is the real root of the equation
(25)2(1-α)r3+(5α-4)r2+(1-3α)r-2α(1-r)2log⁡⁡(1-r)=0
in the interval (0,1).Proof.
Sincef=h+g¯∈Hsp0, we obtain
(26)|bn|≤1n∑k=1n-1k|ak|,(n≥2;a1=1),
by applying Theorem 3. We will make use of (26) to obtain the coefficient bounds for bn in three different cases specified in the theorem. For r∈(0,1), let fr:D→C be defined by
(27)fr(z):=f(rz)r=z+∑n=2∞anrn-1zn+∑n=2∞bnrn-1zn-.
We will show that fr∈FSH*(α). In view of Lemma 6(i), it suffices to show that the sum
(28)S=∑n=2∞n-α1-α|an|rn-1+∑n=2∞n+α1-α|bn|rn-1
is bounded above by 1 for 0≤r<ri for i=1,2,3.
(i) Since|an|≤n, it is easy to deduce that |bn|≤(n-1)(2n-1)/6 by (26). Using these coefficient bounds in (28) and simplifying, we have
(29)S≤16(1-α)[2∑n=2∞n3rn-1+(3+2α)∑n=2∞n2rn-1≤16(1-α)0.+(1-9α)∑n=2∞nrn-1+αr1-r].
Thus S≤1 if r satisfy the inequality
(30)2∑n=2∞n3rn-1+(2α+3)∑n=2∞n2rn-1+(1-9α)∑n=2∞nrn-1+αr1-r≤6(1-α).
By using the identities
(31)r(1-r)2=∑n=1∞nrn,r(1+r)(1-r)3=∑n=1∞n2rn,r(r2+4r+1)(1-r)4=∑n=1∞n3rn
the last inequality reduces to
(32)2(r2+4r+1)(1-r)4+(2α+3)1+r(1-r)3+1-9α(1-r)2+α1-r≤12(1-α)
or equivalently
(33)2(r2+4r+1)+(2α+3)(1-r2)+(1-9α)(1-r)2+α(1-r)3≤12(1-α)(1-r)4.
This gives
(34)12(1-α)r4+(49α-48)r3+8(9-8α)r2+3(11α-18)r+6(1-α)≥0.
Thus by Lemma 6(i), fr∈FSH*(α) for r≤r1 where r1 is the real root of (23) in (0,1). In particular, f is univalent and fully starlike of order α in |z|<r1.
(ii) If|an|≤1 then (26) gives |bn|≤(n-1)/2. These coefficient bounds lead to the following inequality for the sum (28):
(35)S≤12(1-α)×[∑n=2∞n2rn-1+(1+α)∑n=2∞nrn-1-3αr1-r].
Therefore it follows that S≤1 if r satisfy the inequality
(36)∑n=2∞n2rn-1+(1+α)∑n=2∞nrn-1-3αr1-r≤2(1-α).
Making use of identities (31) in the last inequality, we obtain
(37)1+r(1-r)3+1+α(1-r)2-3α1-r≤4(1-α),
which simplifies to
(38)2(1-α)+(7α-12)r+(12-9α)r2-4(1-α)r3≥0.
Lemma 6(i) shows that fr∈FSH*(α) for r≤r2 where r2 is the real root of (24) in (0,1). In particular, f is univalent and fully starlike of order α in |z|<r2.
(iii) Using (26), it is easily seen that |bn|≤(n-1)/n. Using the coefficient bounds for |an| and |bn| in (28), it follows that
(39)S≤11-α[∑n=2∞nrn-1-2α∑n=2∞1nrn-1+αr1-r].
The sum S≤1 if r satisfy the inequality
(40)∑n=2∞nrn-1-2α∑n=2∞1nrn-1+αr1-r≤1-α.
Using (31) and the identity -log⁡⁡(1-r)=r+r2/2+r3/3+⋯, the last inequality reduces to
(41)1(1-r)2+2αrlog⁡⁡(1-r)+α1-r≤2(1-α),
which is equivalent to
(42)2(1-α)r3+(5α-4)r2+(1-3α)r-2α(1-r)2log⁡⁡(1-r)≥0.
By applying Lemma 6(i), fr∈FSH*(α) for r≤r3 where r3 is the real root of (25) in (0,1). In particular, f is univalent and fully starlike of order α in |z|<r3. This completes the proof of the theorem.Remark 8.
By (26), it follows that |b2|≤1/2 for all functions f∈Hsp0. The bound 1/2 is sharp for the function f0(z)=z+z¯2/2∈Hsp0. Since f0 is univalent in D, the coefficient inequality |b2|≤1/2 remains sharp in the subclass SH0. Clunie and Sheil-Small [14] were the first to observe the sharp inequality |b2|≤1/2 for functions in the class SH0.Remark 9.
Letf=h+g¯∈Hsp0, where h and g are given by (9). In the proof of part (i) of Theorem 7, we noticed that if |an|≤n then |bn|≤(n-1)(2n-1)/6. The bound for |bn| coincides with conjectured bound for |bn| when f∈SH0 proposed by Clunie and Sheil-Small [14].The next theorem calculates the radius of full convexity of orderα   (0≤α<1) for the class Hsp0 under certain choices of the coefficient bound on the analytic part.Theorem 10.
Letf=h+g¯∈Hsp0, where h and g are given by (9) with b1=g′(0)=0 and 0≤α<1. Then we have the following. (a)
If|an|≤n or, in particular, h is univalent, then f is fully convex of order α in the disk |z|<s1 where s1=s1(α) is the real root of the equation
(43)2(1-α)r5-10(1-α)r4+2(10-11α)r3+3(7α-6)r2+(15-8α)r-(1-α)=0
in the interval (0,1).(b)
If|an|≤1 or, in particular, h is convex univalent, then f is fully convex of order α in the disk |z|<s2 where s2=s2(α) is the real root of the equation
(44)2(1-α)r4-8(1-α)r3+2(6-5α)r2-5(1-α)r+(1-α)=0
in the interval (0,1).(c)
If|an|≤1/n or, in particular, Reh′(z)>0, then f is fully convex of order α in the disk |z|<s3 where s3=s3(α) is the real root of the equation
(45)2(1-α)r3-2(3-2α)r2+(7-3α)r-(1-α)=0
in the interval (0,1).Proof.
Following the method of the proof of Theorem7, it suffices to show that the function fr defined by (27) belongs to FKH(α). Using the coefficient bounds |an|≤n and |bn|≤(n-1)(2n-1)/6, we deduce that
(46)S′=∑n=2∞n(n-α)1-α|an|rn-1+∑n=2∞n(n+α)1-α|bn|rn-1≤16(1-α)[2∑n=2∞n4rn-1+(2α+3)∑n=2∞n3rn-1≤16(1-α)0.+(1-9α)∑n=2∞n2rn-1+α∑n=2∞nrn-1].
According to Lemma 6(ii), we need to show that S′≤1, or equivalently
(47)2∑n=2∞‍n4rn-1+(2α+3)∑n=2∞n3rn-1+(1-9α)∑n=2∞n2rn-1+α∑n=2∞nrn-1≤6(1-α).
Using (31) and the identity ∑n=1∞n4rn=r(1+r)(1+10r+r2)/(1-r)5, the last inequality reduces to
(48)(1-α)+(8α-15)r+(18-21α)r2+(22α-20)r3+10(1-α)r4-2(1-α)r5≥0.
Lemma 6(ii) shows that fr∈FKH(α) for r≤s1 where s1 is the real root of (43) in (0,1). In particular, f is fully convex of order α in |z|<s1. This proves (a). The proof of (b) and (c) follows on similar lines.The sharpness of the radii constants for the classHsp0 obtained in Theorems 7 and 10 is still unresolved. However, these constants can be shown to be sharp for certain subclasses of H0 as seen by the following theorem.Theorem 11.
LetAn,Bn≥0   (n=2,3,…) and let F be a family of harmonic functions f=h+g¯∈H0 where h and g, given by (9) with b1=g′(0)=0, satisfy |an|≤An and |bn|≤Bn  for n=2,3,…. Furthermore, if rSH(F), rFSH*(α)(F), and rFKH(α)(F) denote, respectively, the radii of univalence, full starlikeness of order α (0≤α<1), and full convexity of order α (0≤α<1) in F, then we have the following.(1)
IfAn=n and Bn=(n-1)(2n-1)/6, then rSH(F)=r1(0)≃0.132529, rFSH*(α)(F)=r1(α), and rFKH(α)(F)=s1(α) where r1=r1(α) and s1=s1(α) are the real roots of (23) and (43), respectively, in (0,1).(2)
IfAn=1 and Bn=(n-1)/2, then rSH(F)=r2(0)=1-1/21/3≃0.206299, rFSH*(α)(F)=r2(α), and rFKH(α)(F)=s2(α) where r2=r2(α) and s2=s2(α) are the real roots of (24) and (44), respectively, in (0,1).(3)
IfAn=1/n and Bn=(n-1)/n, then rSH(F)=r3(0)=1-1/2≃0.292893, rFSH*(α)(F)=r3(α), and rFKH(α)(F)=s3(α) where r3=r3(α) and s3=s3(α) are the real roots of (25) and (45), respectively, in (0,1).Proof.
Note that the roots of (23) in (0,1) are decreasing as functions of α∈[0,1). Consequently, r1(α)≤r1(0). A similar remark holds for (24), (25), and (43)–(45). This observation together with Theorems 7 and 10 gives rSH(F)≥ri(0), rFSH*(α)(F)≥ri(α), and rFKH(α)(F)≥si(α) for i=1,2,3 in the respective three cases specified in the theorem. Therefore it is enough to show that these radii constants are best possible.
(1) For sharpness of the numbersr1(α), let fS:D→C be defined by
(49)fS(z)=2z-z(1-z)2+3z2+z36(1-z)3¯=z-∑n=2∞nzn+16∑n=2∞(n-1)(2n-1)zn¯.
As fS has real coefficients, for r∈(0,1), the Jacobian of fS takes the form
(50)JfS(r)=(1-7r+14r2-8r3+2r4)(1-9r+12r2-8r3+2r4)(1-r)8.
Since JfS(r1(0))=0 the function fS is not univalent in |z|<r if r>r1(0). Also, since
(51)∂∂θarg⁡fS(reiθ)|θ=0,r=r1=6(1-9r1+12r12-8r13+2r14)6-33r1+64r12-49r13+12r14=α
it follows that fS is not fully starlike of order α in |z|<r if r>r1, where r1=r1(α) is the real root of (23) in (0,1).
For sharpness of the numberss1(α), consider the function
(52)fC(z)=2z-z(1-z)2-3z2+z36(1-z)3¯=z-∑n=2∞nzn-16∑n=2∞(n-1)(2n-1)zn¯,
and observe that
(53)∂∂θ(arg⁡{∂∂θfC(reiθ)})|θ=0,r=s1=1-15s1+18s12-20s13+10s14-2s15(1-s1)(1-7s1+14s12-8s13+2s14)=α.
This shows that fC is not fully convex of order α in |z|<r if r>s1, where s1=s1(α) is the real root of (43) in (0,1).
(2) The Jacobian of the functionfS:D→C defined by
(54)fS(z)=2z-z1-z+z22(1-z)2¯=z-∑n=2∞zn+12∑n=2∞(n-1)zn¯
vanishes at z=r2(0) and
(55)∂∂θarg⁡fS(reiθ)|θ=0,r=r2=2(1-6r2+6r22-2r23)2-7r2+9r22-4r23=α.
These two observations imply that the numbers r2(α) are sharp, where r2=r2(α) is the real root of (24) in (0,1). For sharpness of the constants s2(α), observe that the function
(56)fC(z)=2z-z1-z-z22(1-z)2¯=z-∑n=2∞zn-12∑n=2∞(n-1)zn¯
satisfies
(57)∂∂θ(arg⁡{∂∂θfC(reiθ)})|θ=0,r=s2=1-10s2+12s22-8s23+2s241-5s2+10s22-8s23+2s24=α,
where s2=s2(α) is the real root of (44) in (0,1).
(3) The functionfS:D→C defined by
(58)fS(z)=2z+log⁡(1-z)+z1-z+log⁡⁡(1-z)¯=z-∑n=2∞1nzn+∑n=2∞n-1nzn¯
satisfies JfS(r3(0))=0 and
(59)∂∂θarg⁡fS(reiθ)|θ=0,r=r3=r3(1-4r3+2r32)(1-r3)[r3(3-2r3)+2(1-r3)log⁡(1-r3)]=α,
where r3=r3(α) is the real root of (25) in (0,1). If s3=s3(α) is the real root of (45) in (0,1), then
(60)∂∂θ(arg{∂∂θfC(reiθ)})|θ=0,r=s3=1-7s3+6s32-2s331-3s3+4s32-2s33=α,
where fC:D→C is defined by
(61)fC(z)=2z+log⁡(1-z)-(z1-z+log⁡⁡(1-z))¯=z-∑n=2∞1nzn-∑n=2∞n-1nzn-.Now, we will discuss a particular case under which the results of Theorems7 and 10 can be further improved.Theorem 12.
Letf=h+g¯∈Hsp0, where h and g are given by (9) with b1=g′(0)=0. Further, suppose that the dilatation w(z)=g′(z)/h′(z)=z for all z∈D. Then we have the following. (i)
If|an|≤n or, in particular, h is univalent, then f is univalent and fully starlike in the disk |z|<R1 where R1≃0.135918 is the real root of the equation 2r3-5r2+8r-1=0 in the interval (0,1). Moreover, f is fully convex in |z|<S1 where S1≃0.0739351 in the real root of the equation 2r4-8r3+7r2-14r+1=0 in the interval (0,1).(ii)
If|an|≤1 or, in particular, h is convex univalent, then f is univalent and fully starlike in the disk |z|<R2 where R2=(5-17)/4≃0.219224. Also, f is fully convex in |z|<S2 where S2≃0.120385 in the real root of the equation 2r3-6r2+9r-1=0 in the interval (0,1).(iii)
If|an|≤1/n or, in particular, Reh′(z)>0, then f is univalent and fully starlike in the disk |z|<R3 where R3=1/3≃0.333333. And f is fully convex in |z|<S3 where S3=(3-6)/3≃0.183503.Proof.
Settingw1=1 and wn=0 for n≠1 in (18), we obtain nbn=(n-1)an-1 so that
(62)|bn|≤n-1n|an-1|(n≥2;a1=1).
Let fr be defined by (27). For the proof of (i), note that since |an|≤n, it is easily seen that |bn|≤(n-1)2/n using (62). Using these coefficient bounds, we have
(63)S=∑n=2∞n|an|rn-1+∑n=2∞n|bn|rn-1≤∑n=2∞n2rn-1+∑n=2∞(n-1)2rn-1=(1+r)2(1-r)3-1,
using the identities (31). Thus S≤1 if r satisfy the inequality 2(1-r)3≥(1+r)2 or 1-8r+5r2-2r3≥0. By Lemma 6(i), it follows that fr∈FSH* for r≤R1 where R1 is the real root of 2r3-5r2+8r-1=0 in (0,1). In particular, f is univalent and fully starlike in |z|<R1. For full convexity, observe that
(64)S′=∑n=2∞n2|an|rn-1+∑n=2∞n2|bn|rn-1≤∑n=2∞n3rn-1+∑n=2∞n(n-1)2rn-1=5r2+6r+1(1-r)4-1.
The sum S′≤1 if r satisfy the inequality 2r4-8r3+7r2-14r+1≥0. Thus Lemma 6(ii) shows that fr∈FKH for r≤S1 where S1 is the real root of 2r4-8r3+7r2-14r+1=0 in (0,1). In particular, f is fully convex in |z|<S1. This proves (i). The other two parts of the theorem are similar and hence their proofs are omitted.Remark 13.
Observe thatri(0)<Ri   (i=1,2,3) and si(0)<Si   (i=1,2,3). Here ri(0), si(0), Ri, and Si are as defined in Theorems 7, 10, and 12.Remark 14.
Iff=h+g¯∈Hsp0, where h and g are given by (9) with b1=g′(0)=0 and the dilatation w(z)=g′(z)/h′(z) is given by w(z)=zm (m≥1), then
(65)|bn|≤n-mn|an-m|(n≥m+1;a1=1),
by setting wm=1 and wn=0 for n≠m in (18). Radius constants may be obtained in this case by carrying out a similar calculation as in the proof of Theorem 12.


## 3. Sufficient Coefficient Estimates for Full Starlikeness and Convexity

In this section, we determine sufficient coefficient inequalities for functions to be in the classesFSH*(α) and FKH(α). As an application, these results are applied to hypergeometric functions in Section 4.Theorem 15.
Letf=h+g¯∈H, where h and g are given by (9). Suppose that λ∈(0,1]. Then one has the following.(a)
If(66)∑n=2∞n|an|+∑n=1∞n|bn|≤λ,
then f is fully starlike of order 2(1-λ)/(2+|b1|+λ).(b)
If(67)∑n=2∞n2|an|+∑n=1∞n2|bn|≤λ,
then f is fully starlike of order 2(2-λ-|b1|)/(4+3|b1|+λ). Moreover, f is fully convex of order 2(1-λ)/(2+|b1|+λ).The results are sharp.Proof.
If we setα0=2(1-λ)/(2+|b1|+λ) then α0∈[0,1) and
(68)∑n=2∞n-α01-α0|an|+∑n=1∞n+α01-α0|bn|≤∑n=2∞n+α01-α0|an|+∑n=1∞n+α01-α0|bn|=11-α0(∑n=2∞‍n|an|+∑n=1∞‍n|bn|)+α01-α0(∑n=2∞|an|+∑n=1∞|bn|)≤λ1-α0+α01-α0(12∑n=2∞n(|an|+|bn|)+|b1|)≤λ1-α0+α01-α0(λ-|b1|2+|b1|)=2λ+α0(λ+|b1|)2(1-α0)=1.
By Lemma 6(i), it follows that f is fully starlike of order 2(1-λ)/(2+|b1|+λ). The harmonic function
(69)f1(z)=z+|b1|z¯+λ-|b1|2z¯2,|b1|<λ,
satisfies the coefficient inequality (66). Further, for z=reiθ, we have
(70)∂∂θargf1(reiθ)=Re(2(z-|b1|z¯-(λ-|b1|)z¯2)2(z+|b1|z¯)+(λ-|b1|)z¯2)≥2(1-|b1|-(λ-|b1|)|z|)2(1+|b1|)+(λ-|b1|)|z|>2(1-λ)2+|b1|+λ
which shows that the bound for the order of full starlikeness is sharp. This proves (a).
For the proof of (b), observe that(71)∑n=2∞n|an|+∑n=1∞n|bn|≤12∑n=2∞n2(|an|+|bn|)+|b1|≤12(λ-|b1|)+|b1|=λ+|b1|2:=μ0(say),
using (67). Since μ0∈(0,1), f is fully starlike of order 2(1-μ0)/(2+|b1|+μ0)=2(2-λ-|b1|)/(4+3|b1|+λ) by part (a) of the theorem. For the order of full convexity of f, note that
(72)∑n=2∞n(n-α0)1-α0|an|+∑n=1∞n(n+α0)1-α0|bn|≤∑n=2∞n(n+α0)1-α0|an|+∑n=1∞n(n+α0)1-α0|bn|=11-α0(∑n=2∞n2|an|+∑n=1∞n2|bn|)+α01-α0(∑n=2∞n|an|+∑n=1∞n|bn|)≤λ1-α0+α01-α0(λ+|b1|2)=2λ+α0(λ+|b1|)2(1-α0)=1,
where α0 is as defined in the proof of part (a) of the theorem. By Lemma 6(ii), f is fully convex of order 2(1-λ)/(2+|b1|+λ). In this case, the harmonic function
(73)f2(z)=z+|b1|z¯+λ-|b1|4z¯2,|b1|<λ,
shows that the result is best possible.Ifb1=0, then Theorem 15 reduces to [26, Theorem 3.6 and Corollary 3.7]. Also, Theorem 15 gives the following two corollaries.Corollary 16.
Letf=h+g¯∈H, where h and g are given by (9) and 0≤α<2/(2+|b1|). Then we have the following.(i)
If(74)∑n=2∞n|an|+∑n=1∞n|bn|≤2-(2+|b1|)α2+α,
then f∈FSH*(α).(ii)
If(75)∑n=2∞n2|an|+∑n=1∞n2|bn|≤2-(2+|b1|)α2+α,
then f∈FKH(α).All these results are sharp.Proof.
First, we will prove (i). Settingλ0=(2-(2+|b1|)α)/(2+α) we see that λ0∈(0,1] and the coefficient inequality (66) is satisfied for λ=λ0. Hence by Theorem 15(a), f is fully starlike of order 2(1-λ0)/(2+|b1|+λ0)=α. This proves (i). For part (ii), since inequality (67) is satisfied for λ=λ0 it follows that f is fully convex of order 2(1-λ0)/(2+|b1|+λ0)=α by Theorem 15(b). The functions
(76)f1(z)=z+|b1|z¯+(1-|b1|)-(1+|b1|)α2+αz¯2,f2(z)=z+|b1|z¯+(1-|b1|)-(1+|b1|)α2(2+α)z¯2
show that the upper bound (2-(2+|b1|)α)/(2+α) is best possible in (i) and (ii), respectively.Corollary 17.
Letf=h+g¯∈H, where h and g are given by (9) and α∈R satisfies
(77)2(1-|b1|)5+3|b1|≤α<2(2-|b1|)4+3|b1|.
If
(78)∑n=2∞‍n2|an|+∑n=1∞n2|bn|≤2(2-|b1|)-α(4+3|b1|)2+α,
then f∈FSH*(α). The function
(79)f0(z)=z+|b1|z¯+(1-|b1|)-(1+|b1|)α2+αz¯2
shows that the bound (2(2-|b1|)-α(4+3|b1|))/(2+α) is best possible.Proof.
If we setν0=(2(2-|b1|)-α(4+3|b1|))/(2+α), then ν0∈(0,1] and the coefficient inequality (67) is satisfied for λ=ν0 using the hypothesis. By Theorem 15(b), f is fully starlike of order 2(2-ν0-|b1|)/(4+3|b1|+ν0)=α as desired.Ifb1=0, then Corollaries 16 and 17 reduce to the following theorem.Theorem 18.
Letf=h+g¯∈H, where h and g are given by (9) with b1=g′(0)=0 and let α∈R.(1)
Ifα∈[0,1), then the sharp implications hold:
(80)∑n=2∞n(|an|+|bn|)≤2(1-α)2+α⟹f∈FSH*(α),∑n=2∞n2(|an|+|bn|)≤2(1-α)2+α⟹f∈FKH(α).(2)
Ifα∈[2/5,1), then
(81)∑n=2∞n2(|an|+|bn|)≤4(1-α)2+α⟹f∈FSH*(α).
In particular, we have
(82)∑n=2∞n2(|an|+|bn|)≤1⟹f∈FSH*(25).


## 4. Interplay between Hypergeometric Functions and Full Starlikeness and Convexity

In recent years, there has been a growth of interest in the interplay between hypergeometric functions and harmonic mappings inD; see [27–31]. Let F(β,γ,δ;z) be the Gaussian hypergeometric function defined by
(83)F(β,γ,δ;z):=∑n=0∞(β)n(γ)n(δ)n(1)nzn,z∈D,
where β,γ,δ are complex numbers with δ≠0,-1,-2,…, and (θ)n is the Pochhammer symbol: (θ)0=1 and (θ)n=θ(θ+1)⋯(θ+n-1) for n=1,2,…. Since the hypergeometric series in (83) converges absolutely in D, it follows that F(β,γ,δ;z) defines an analytic function in D and plays an important role in the theory of univalent functions.The first author and Silverman [27] initiated the study of harmonic functions ϕ1+ϕ¯2 where ϕ1(z)≡ϕ1(β1,γ1,δ1;z) and ϕ2(z)≡ϕ2(β2,γ2,δ2;z) are the hypergeometric functions defined by
(84)ϕ1(z):=zF(β1,γ1,δ1;z),ϕ2(z):=F(β2,γ2,δ2;z)-1.
Making use of Corollaries 16 and 17, we determine the sufficient conditions in terms of hypergeometric inequalities for the function Φ=ϕ1+ϕ¯2 to be in the classes FSH*(α) and FKH(α). However, we first need the well-known Gauss summation formula
(85)F(β,γ,δ;z)=Γ(δ)Γ(δ-β-γ)Γ(δ-β)Γ(δ-γ),Re(δ-β-γ)>0
and the following result by the first author [29].Lemma 19.
Ifβ,γ,δ>0, then(i)
F(β+k,γ+k,δ+k;1)=((δ)k/(δ-β-γ-k)k)F(β,γ,δ;1) for k=0,1,2,…, if δ>β+γ+k;(ii)
∑n=2∞‍(n-1)((β)n-1(γ)n-1/(δ)n-1(1)n-1)=(βγ/(γ-β-γ-1))F(β,γ,δ;1) if δ>β+γ+1;(iii)
∑n=2∞‍(n-1)2((β)n-1(γ)n-1/(δ)n-1(1)n-1)=((β)2(γ)2/(γ-β-γ-1)2+βγ/(γ-β-γ-1))F(β,γ,δ;1) if δ>β+γ+2.Theorem 20.
Letβj,γj∈C, and δj∈R satisfy δj>|βj|+|γj|+1 for j=1,2. Set η=β2γ2/δ2 and let 0≤α<2/(2+|η|). If
(86)(|β1||γ1|δ1-|β1|-|γ1|-1+1)F(|β1|,|γ1|,δ1;1)+|β2||γ2|δ2-|β2|-|γ2|-1F(|β2|,|γ2|,δ2;1)≤4-(1+|η|)α2+α
then Φ=ϕ1+ϕ¯2∈FSH*(α), where ϕ1 and ϕ2 are given by (84).Proof.
Observe that(87)Φ(z)=z+∑n=2∞(β1)n-1(γ1)n-1(δ1)n-1(1)n-1zn+∑n=1∞(β2)n(γ2)n(δ2)n(1)nzn¯.
Using the fact |(θ)n|≤(|θ|)n, Gauss summation formula given by (85), and Lemma 19, we have
(88)∑n=2∞n|(β1)n-1(γ1)n-1(δ1)n-1(1)n-1|+∑n=1∞n|(β2)n(γ2)n(δ2)n(1)n|≤∑n=2∞n(|β1|)n-1(|γ1|)n-1(δ1)n-1(1)n-1+∑n=1∞n(|β2|)n(|γ2|)n(δ2)n(1)n=∑n=1∞(n+1)(|β1|)n(|γ1|)n(δ1)n(1)n+∑n=1∞n(|β2|)n(|γ2|)n(δ2)n(1)n=(|β1||γ1|δ1-|β1|-|γ1|-1+1)F(|β1|,|γ1|,δ1;1)+|β2||γ2|δ2-|β2|-|γ2|-1F(|β2|,|γ2|,δ2;1)-1≤4-(1+|η|)α2+α-1=2-(2+|η|)α2+α.
By Corollary 16(i), Φ∈FSH*(α).Theorem 21.
Letβj,γj∈C, and δj∈R satisfy δj>|βj|+|γj|+2 for j=1,2. Set η=β2γ2/δ2. Then one has the following.(i)
If0≤α<2/(2+|η|) and
(89)((|β1|)2(|γ1|)2(δ1-|β1|-|γ1|-2)2+3|β1||γ1|δ1-|β1|-|γ1|-1+1)×F(|β1|,|γ1|,δ1;1)+((|β2|)2(|γ2|)2(δ2-|β2|-|γ2|-2)2+|β2||γ2|δ2-|β2|-|γ2|-1)×F(|β2|,|γ2|,δ2;1)≤4-(1+|η|)α2+α
then Φ=ϕ1+ϕ¯2∈FKH(α).(ii)
If2(1-|η|)/(5+3|η|)≤α<2(2-|η|)/(4+3|η|) and
(90)((|β1|)2(|γ1|)2(δ1-|β1|-|γ1|-2)2+3|β1||γ1|δ1-|β1|-|γ1|-1+1)×F(|β1|,|γ1|,δ1;1)+((|β2|)2(|γ2|)2(δ2-|β2|-|γ2|-2)2+|β2||γ2|δ2-|β2|-|γ2|-1)×F(|β2|,|γ2|,δ2;1)≤3(2-α)-|η|(2+3α)2+α
then Φ=ϕ1+ϕ¯2∈FSH*(α), where ϕ1 and ϕ2 are given by (84).Proof.
Note that(91)∑n=2∞n2|(β1)n-1(γ1)n-1(δ1)n-1(1)n-1|+∑n=1∞n2|(β2)n(γ2)n(δ2)n(1)n|≤∑n=2∞n2(|β1|)n-1(|γ1|)n-1(δ1)n-1(1)n-1+∑n=1∞n2(|β2|)n(|γ2|)n(δ2)n(1)n=∑n=1∞(n+1)2(β1)n(γ1)n(δ1)n(1)n+∑n=1∞n2(|β2|)n(|γ2|)n(δ2)n(1)n=((|β1|)2(|γ1|)2(δ1-|β1|-|γ1|-2)2+3|β1||γ1|δ1-|β1|-|γ1|-1+1)×F(|β1|,|γ1|,δ1;1)+((|β2|)2(|γ2|)2(δ2-|β2|-|γ2|-2)2+|β2||γ2|δ2-|β2|-|γ2|-1)×F(|β2|,|γ2|,δ2;1)-1.
Under the hypothesis of part (i), it is easy to see that
(92)∑n=2∞n2|(β1)n-1(γ1)n-1(δ1)n-1(1)n-1|+∑n=1∞n2|(β2)n(γ2)n(δ2)n(1)n|≤4-(1+|η|)α2+α-1=2-(2+|η|)α2+α.
By Corollary 16(ii), it follows that Φ=ϕ1+ϕ¯2∈FKH(α). Hypothesis of part (ii) shows
(93)∑n=2∞n2|(β1)n-1(γ1)n-1(δ1)n-1(1)n-1|+∑n=1∞n2|(β2)n(γ2)n(δ2)n(1)n|≤3(2-α)-|η|(2+3α)2+α-1=2(2-|η|)-α(4+3|η|)2+α.
Hence Φ∈FSH*(α) by Corollary 17.



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*Source: 454152-2014-09-09.xml*