Radius of Starlikeness for Analytic Functions with Fixed Second Coefficient

Rosihan M. Ali<br />Virendra Kumar<br />V. Ravichandran<br />Shanmugam Sivaprasad Kumar

doi:10.5666/KMJ.2017.57.3.473

Article

Kyungpook Mathematical Journal 2017; 57(3): 473-492

Published online September 23, 2017

Radius of Starlikeness for Analytic Functions with Fixed Second Coefficient

Rosihan M. Ali¹
Virendra Kumar²
V. Ravichandran³
Shanmugam Sivaprasad Kumar⁴

School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia¹
Department of Mathematics, Central University of Haryana, Mahendergarh--123029 Haryana, India²
Department of Mathematics, University of Delhi, Delhi--110007, India³
Department of Applied Mathematics, Delhi Technological University, Delhi--110042, India⁴

Received: August 20, 2015; Accepted: February 26, 2016

Abstract

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Abstract
1. Introduction
2. Preliminaries
3. Radius constants
Acknowledgements
References

Sharp radius constants for certain classes of normalized analytic functions with fixed second coefficient, to be in the classes of starlike functions of positive order, parabolic starlike functions, and Sokół-Stankiewicz starlike functions are obtained. Our results extend several earlier works.

Keywords: Starlike functions, Sokó,l-Stankiewicz starlike functions, parabolic starlike functions, convex functions, radius constants

1. Introduction

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Abstract
1. Introduction
2. Preliminaries
3. Radius constants
Acknowledgements
References

Let denote the class of analytic functions f defined on := {z ∈ ℂ : |z| < 1}, which are normalized by the conditions f(0) = 0, and f′(0) = 1 and let denote its subclass consisting of univalent functions. The well-known Bieberbach theorem states that the second coefficient in the Maclaurin series of functions in is bounded by two. This estimate for the second coefficient plays an important role in the study of the class , and for that reason, there has been considerable continued interest in the investigation of the class ⊂ of functions f(z) = z + a₂z² + · · ·, a₂ = 2b for a fixed b with |b| ≤ 1. The investigation on was initiated as early as 1920 by Gronwall [7], where growth and distortion estimates were obtained for functions in . Recently, Ali et al. [5] extended the theory of second-order differential subordination to the class of analytic functions with fixed second coefficient. Pursuant to that work, Nagpal and Ravichandran [15] obtained sufficient conditions for starlikeness and close-to-convexity. Differential superordinations were considered by Mendiratta et al. [13, 14], while Lee et al. [9] investigated other applications of differential subordination for functions with fixed second coefficient. Livingston problems for close-to-convex functions with fixed second coefficient were studied by Mendiratta and Ravichandran [12]. A survey on functions with fixed initial coefficient can be found in [2]. For 0 ≤ α < 1, the classes (α) and (α) of starlike functions of orderα and convex functions of orderα consist of functions f ∈ satisfying respectively Re (zf′(z)/f(z)) > α, and Re (1 + zf″(z)/f′(z)) > α; the classes := (0) and := (0) are the familiar classes of starlike and convex functions respectively. The second coefficient of functions in these classes satisfies respectively the inequalities |a₂| ≤ 2(1 − α) and |a₂| ≤ 1 − α. For notational convenience, let us denote by , the class of normalized analytic functions of the form f(z) = z + bz₂ + · · ·. For |b| ≤ 1 and 0 ≤ α < 1, let Sb*(α):=S*(α)∩A2b(1-α) and (α) := (α)∩ . Functions in these classes are respectively called starlike and convex functions of order α with fixed second coefficient. Let Sb*:=Sb*(0) and := (0). The class SL* of Sokół-Stankiewicz starlike functions [22] consists of functions f ∈ for which zf′(z)/f(z) lies in the region bounded by the right half-plane of the lemniscate of Bernoulli: |w² − 1| = 1. A function f ∈ is uniformly convex if and only if Re(1+zf″(z)/f′(z)) > |zf″(z)/f′(z)|. The corresponding class of starlike functions connected with the Alexander relation is the class of parabolic starlike functions, introduced by Rønning [19], given by

SP*:={f∈A:Re (zf′(z)f(z))>|zf′(z)f(z)-1|}.

For a survey of uniformly starlike/convex functions, see [1]. For β > 1, the class ℳ(β) consists of functions f ∈ satisfying Re(zf′(z)/f(z)) < β. This class contains non-univalent functions and was investigated in [17, 24] (see also [4]). Clearly, SL*⊂S*,SP*⊂S*(1/2) while ℳ(β) ⊄ .

The classes of starlike, convex and several other functions are related to the class ℘(α), of analytic functions p(z) = 1 + b₁z + b₂z² + · · · satisfying Re(p(z)) > α (0 ≤ α < 1), ℘ := ℘(0). It is well known [16, p. 170] that |b_n| ≤ 2(1 − α) for p ∈ ℘(α). We shall denote by ℘_b(α) the subclass of ℘(α) consisting of functions of the form p(z) = 1 + 2b(1 − α)z + · · ·, |b| ≤ 1, and let ℘_b := ℘_b(0).

Given two sub-families S₁ and S₂ of , the S₁-radius of S₂ is defined to be the largest number ρ such that r⁻¹f(rz) ∈ S₁ for all 0 < r ≤ ρ and for all f ∈ S₂. Several works on radius problems can be found in [18, 21, 23]. In a recent paper, Ali et al. [4] obtained sharp radius estimates for functions f ∈ satisfying certain conditions on the ratio f/g for a given g ∈ . The radii results presented here are nice extensions of Ali et al. [4] and the works of [2, 18, 21, 23] for functions with fixed second coefficient, and include the radii results for the classes of starlike functions of positive order, parabolic starlike functions, and the Sokół-Stankiewicz starlike functions.

2. Preliminaries

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Abstract
1. Introduction
2. Preliminaries
3. Radius constants
Acknowledgements
References

The results that are required in the present investigation are enlisted below:

Lemma 2.1. ([11, Theorem 2])

Let |b| ≤ 1 and 0 ≤ α < 1. If p ∈ ℘_b(α), then, for |z| = r < 1,

|zp′(z)p(z)|≤2(1-α)r1-r2∣b∣r2+2r+∣b∣(1-2α)r2+2(1-α)∣b∣r+1.

Lemma 2.2. ([10, Lemma 1])

Let |b| ≤ 1 and 0 ≤ α < 1. If p ∈ ℘_b(α), then, for |z| = r < 1, |p(z) − C_b| ≤ D_b, where

Cb=(1+∣b∣r)2+(1-2α)(∣b∣+r)2r2(1+2∣b∣r+r2)(1-r2), Db=2(1-α)(∣b∣+r)(1+∣b∣r)r(1+2∣b∣r+r2)(1-r2).

Lemma 2.3. ([10, Theorem 1])

Let |b| ≤ 1 and 0 ≤ α < 1. Suppose p ∈ ℘_b(α). Then, for |z| = r < 1,

Re (zp′(z)p(z))≥{-2(1-α)(∣b∣+2r+∣b∣r2)r(1+2α∣b∣r+(2α-1)r2)(1+2∣b∣r+r2),R′≤Rb;(2αC1-C1-α)/(1-α)R′≥Rb,

where R_b = C_b − D_b, R′=αC1, C_band D_bare as given in Lemma.

Lemma 2.4. ([5, Theorem 5.1])

If f(z) = z+a₂z²+· · · ∈ , then f ∈ (α), whereαis the smallest positive root of the equation 2α³ − |a₂|α² − 4α + 2 = 0, in the interval [1/2, 2/3].

Lemma 2.5. ([3, Lemma 2.2])

For 0<a<2, let r_abe given by

ra={(1-a2-(1-a2))1/2,0<a≤22/3;2-a,22/3≤a<2,

and for a > 0, let R_abe given by

Ra={2-a,0<a≤1/2;a,1/2≤a.

Then {w : |w − a| < r_a} ⊂ {w : |w² − 1| < 1} ⊂ {w : |w − a| < R_a}.

Lemma 2.6. ([20, Section 3])

Let a > 1/2. If the number R_ais given by

Ra={a-1/2,1/2<a≤3/2;2a-2,a≥3/2,

then {w ∈ ℂ : |w − a| < R_a} ⊂ {w ∈ ℂ : |w − a| < Rew}.

3. Radius constants

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Abstract
1. Introduction
2. Preliminaries
3. Radius constants
Acknowledgements
References

Let

(3.1)f(z)=z+a2z2+⋯

and if Re(f(z)/z) > 0, then f(z)/z ∈ ℘ and hence |a₂| ≤ 2. So such functions can be given the series expansion: f(z) = z + 2bz² + · · ·, where |b| ≤ 1.

Definition 3.1

For |b| ≤ 1, let Fb1 be the class of functions f ∈ such that Re (f(z)/z) > 0.

We now give below the radius constants pertaining to the class Fb1:

Theorem 3.2

The sharp radius constants for the class Fb1 are enlisted below:

The SL*–radius is the smallest positive root r₀ ∈ (0, 1) of
(3.2)(2-1)r4+22∣b∣r3+4r2+2∣b∣(2-2)r-2+1=0,
The ℳ(β)–radius is the smallest positive root r₁ ∈ (0, 1) of
(3.3)(β-1)r4+2∣b∣βr3+4r2+2∣b∣(2-β)r-β+1=0.
The(α)–radius is the smallest positive root r₂ ∈ (0, 1) of
(3.4)(1-α)r4+2∣b∣(2-α)r3+4r2+2∣b∣αr+α-1=0.
The SP*–radius is the smallest positive root r₃ ∈ (0, 1) of
(3.5)r4+6∣b∣r3+8r2+2∣b∣r-1=0.

Proof

Clearly, the function p(z) = f(z)/z = 1 + 2bz + · · · ∈ ℘_b and

zp′(z)p(z)=zf′(z)f(z)-1.

Now by taking α = 0 in Lemma 2.1, we have

(3.6)|zf′(z)f(z)-1|=|zp′(z)p(z)|≤2r(∣b∣r2+2r+∣b∣)(1-r2)(r2+2∣b∣r+1).

(1) From Lemma 2.5, we see that

|(zf′(z)f(z))2-1|<1

whenever the following inequality holds:

2r(∣b∣r2+2r+∣b∣)(1-r2)(r2+2∣b∣r+1)≤2-1,

which upon simplification, becomes

1-2+2∣b∣(2-2) r+4r2+22∣b∣r3+(2-1) r4≤0.

Therefore, the SL*-radius for the class Fb1, is the smallest positive root r₀ ∈ (0, 1) of (3.2).

To prove the sharpness, consider the function f₀ defined by

(3.7)f0(z)=z(1+2bz+z2)1-z2

together with w(z) := z(z + b)/(1 + bz). Then we see that

f0(z)z=1+w(z)1-w(z),

where w is an analytic function satisfying the conditions of Schwarz’s lemma in the unit disk , which leads to Re(f₀(z)/z) > 0 in and hence f0∈Fb1. Thus, for z = r₀, the root of (3.2), we have

z f0′(z)f0(z)=1+4br0+4r02-r04(1-r02) (1+2br0+r02)=2,

it follows that

|(z f0′(z)f0(z))2-1|=1 (z=r0),

which establishes sharpness of the result.

(2) The inequality (3.6) shows that

Re (z f′(z)f(z))≤1+2r(∣b∣r2+2r+∣b∣)(1-r2)(r2+2∣b∣r+1)≤β,

if the following inequality

(β-1)r4+2∣b∣βr3+4r2+2(2-β)∣b∣r+1-β≤0

holds. Therefore, the ℳ(β)-radius of the class Fb1 is the smallest positive root r₁ ∈ (0, 1) of (3.3). The result is sharp due to the function given in (3.7) as, for z = r₁, the root of (3.3), we see that

z f0′(z)f0(z)=1+4br1+4r12-r14(1-r12) (1+2br1+r12)=β.

(3) In view of (3.6), it follows that

Re (z f′(z)f(z))≥1-2r(∣b∣r2+2r+∣b∣)(1-r2)(r2+2∣b∣r+1)≥α,

whenever the following inequality

(1-α)r4+2∣b∣(2-α)r3+4r2+2∣b∣αr+α-1≤0

holds. Thus, the (α)–radius of the class Fb1 is the smallest positive root r₂ ∈ (0, 1) of (3.4).

The function f₀ defined by

(3.8)f0(z)=z(1-z2)1-2bz+z2

is in the class Fb1 because for the function f₀ defined in (3.8), we have f₀(z)/z = (1−w(z))/(1+w(z)), where w(z) = z(z−b)/(1−bz) is an analytic function satisfying the conditions of Schwarz’s lemma in the unit disk , and hence Re(f₀(z)/z) > 0 in . The result is sharp for the function given in (3.8) as, for z = −r₂, the root of (3.4), we have

Re (z f0′(z)f0(z))=z f0′(z)f0(z)=1-r22 (4-4br2+r22)(1-r22) (1-2br2+r22)=α,

which demonstrates sharpness.

(4) Lemma 2.6 shows that the disk (3.6) lies inside the parabolic region Ω = {w : |w − 1| < Rew} provided that

2r(∣b∣r2+2r+∣b∣)(1-r2)(r2+2∣b∣r+1)≤12,

or equivalently, if the inequality r⁴ + 6|b|r³ + 8r² + 2|b|r − 1 ≤ 0 holds. Thus, the SP*–radius of the class Fb1 is the smallest positive root r₃ ∈ (0, 1) of (3.5).

The function defined in (3.8), for z = −r₃ satisfies

z f0′(z)f0(z)=1-r32 (4-4br3+r32)(1-r32) (1-2br3+r32)=12,

which demonstrates sharpness. The following figures illustrate sharpness of the result.

Remark 3.3

For α = 0, part (3) of Theorem 3.1 reduces to the result [8, Theorem 2] of Goel.

Let

(3.9)g(z)=z+g2z2+⋯

and assume that g(z)/z ∈ ℘. Let f be given by (3.1) and Re(f(z)/g(z)) > 0. Then we have |a₂| ≤ |g₂| + 2 ≤ 4. Our next theorem focuses on the class of functions involving these functions f and g with fixed second coefficients, whose series expansions are given respectively by f(z) = z + 4bz² + · · · and g(z) = z + 2cz² + · · ·, where |b| ≤ 1 and |c| ≤ 1.

Definition 3.4

For |b| ≤ 1 and |c| ≤ 1, letFb,c2:={f∈A4b:Re (f(z)g(z))>0 and Re (g(z)z>0, where g∈A2c)}.

Here below, we furnish the radius constants for the class Fb,c2:

Theorem 3.5

Assume thatγ := |2b − c|. Then the sharp radius constants for the class Fb,c2 are enlisted below:

The SL*–radius is the smallest positive root r₀ ∈ (0, 1) of
(3.10)(2-1)r6+(∣c∣+γ)22r5+(7+2+4(1+2)∣c∣γ)r4+12(∣c∣+γ)r3+(9-2+4(3-2)∣c∣γ)r2+2(2-2)(∣c∣+γ)r-2+1=0.
The ℳ(β)–radius is the smallest positive root r₁ ∈ (0, 1) of
(3.11)(β-1)r6+2β(∣c∣+γ)r5+(7+β+4(1+β)∣c∣γ)r4+12(∣c∣+γ)r3+(9-β+4(3-β)∣c∣γ)r2+2(2-β)(∣c∣+γ)r-β+1=0.
The(α)–radius is the smallest positive root r₂ ∈ (0, 1) of
(3.12)(1-α)r6+2(2-α)(∣c∣+γ)r5+(9-α+4(3-α)∣c∣γ)r4+12(∣c∣+γ)r3+(7+α+4(1+α)∣c∣γ)r2+2(∣c∣+γ)αr+α-1=0.
The SP*–radius is the smallest positive root r₃ ∈ (0, 1) of
(3.13)r6+6(∣c∣+γ)r5+(17+20γ∣c∣)r4+24(∣c∣+γ)r3+(15+12γ∣c∣)r2+2(∣c∣+γ)r-1=0.

Proof

Let the functions p and h be defined by p(z) = g(z)/z, and h(z) = f(z)/g(z). Then

p(z)=1+2cz+⋯ and h(z)=1+2(2b-c)z+⋯

or p ∈ ℘_c and h ∈ ℘₂_b₋_c. Since f(z) = zp(z)h(z), from Lemma 2.1 with α = 0, we have

(3.14)|z f′(z)f(z)-1|≤|zp′(z)p(z)|+|zh′(z)h(z)|≤2r1-r2 (∣c∣r2+2r+∣c∣r2+2∣c∣r+1+γr2+2r+γr2+2γr+1).

(1) By Lemma 2.5, the function f satisfies|(zf′(z)/f(z))² − 1| < 1, for |z| < r, if the following inequality holds

2r1-r2 (∣c∣r2+2r+∣c∣r2+2∣c∣r+1+γr2+2r+γr2+2γr+1)≤2-1

or equivalently, if the following inequality holds:

(2-1)r6+(∣c∣+γ)22r5+(7+2+4(1+2)∣c∣γ)r4+12(∣c∣+γ)r3+(9-2+4(3-2)∣c∣γ)r2+2(2-2)(∣c∣+γ)r-2+1≤0.

Therefore, the SL*-radius of the class Fb,c2 is the smallest positive root r₀ ∈ (0, 1) of (3.10).

Consider the functions defined by

(3.15)f0(z)=z (1+(4b-2c)z+z2) (1+2cz+z2)(1-z2)2 and g0(z)=z (1+2cz+z2)(1-z2).

The function f₀ with the choice of g₀, defined above, is in the class Fb,c2 because

f0(z)g0(z)=1+w1(z)1-w1(z) and g0(z)z=1+w2(z)1-w2(z),

where w₁(z) = z(z + 2b − c)/(1 + (2b − c)z) with |2b − c| ≤ 1 and w₂(z) = z(z + c)/(1 + cz) are analytic functions satisfying the conditions of Schwarz’s lemma in the unit disk , and hence Re(g₀(z)/z) > 0 and Re(f₀(z)/g₀(z)) > 0 in . Since

(3.16)z f0′(z)f0(z)=1+21-r0+21+r0-2(1+cr0)1+2cr0+r02-2+4br0-2cr01+r0(4b-2c+r0)=2, (z=r0),

we have

|(zf0(z)f0(z))2-1|=1.

Thus, the result is sharp.

(2) The inequality (3.14) shows that

Re (z f′(z)f(z))≤1+2r1-r2 (∣c∣r2+2r+∣c∣r2+2∣c∣r+1+γr2+2r+γr2+2γr+1)≤β,

if the following inequality holds:

(β-1)r6+2β(∣c∣+γ)r5+(7+β+4(1+β)∣c∣γ)r4+12(∣c∣+γ)r3+(9-β+4(3-β)∣c∣γ)r2+2(2-β)(∣c∣+γ)r-β+1≤0.

Hence the ℳ(β)-radius of the class Fb,c2 is the smallest positive root r₁ ∈ (0, 1) of (3.11). The result is sharp due to the functions given in (3.15) as it can be seen for z = r₁

(3.17)z f0′(z)f0(z)=1+21-r1+21+r1-2(1+cr1)1+2cr1+r12-2+4br1-2cr11+r1(4b-2c+r1)=β.

(3) In view of (3.14), it follows that

Re (z f′(z)f(z))≥1-2r1-r2 (∣c∣r2+2r+∣c∣r2+2∣c∣r+1+γr2+2r+γr2+2γr+1)≥α,

if the following inequality holds:

(1-α)r6+2(2-α)(∣c∣+γ)r5+(9-α+4(3-α)∣c∣γ)r4+12(∣c∣+γ)r3+(7+α+4(1+α)∣c∣γ)r2+2(∣c∣+γ)αr+α-1≤0.

Thus, the (α)–radius of the class Fb,c2 is the smallest positive root r₂ ∈ (0, 1) of (3.12).

Consider the functions defined by

(3.18)f0(z)=z (1-z2)2(1-(4b-2c)z+z2) (1-2cz+z2) and g0(z)=z (1-z2)(1-2cz+z2).

The function f₀, with the choice of g₀ defined in (3.18), is in the class Fb2 because

f0(z)g0(z)=1-w1(z)1+w1(z) and g0(z)z=1-w2(z)1+w2(z),

where w₁(z) = z(z − (2b − c))/(1 − (2b − c)z) with |2b − c| ≤ 1 and w₂(z) = z(z−c)/(1−cz) are analytic functions satisfying the conditions of Schwarz’s lemma in the unit disk , and hence Re(g₀(z)/z) > 0 and Re(f₀(z)/g₀(z)) > 0 in . The functions defined in (3.18) satisfy

z f0′(z)f0(z)=1-21+r2-21-r2+2+2cr21+2cr2+r22+2(1+2br2-cr2)1+r2(4b-2c+r2)=α (z=-r2),

which demonstrates the sharpness.

(4) By Lemma 2.6, the disk (3.14) lies inside the parabolic region Ω = {w : |w − 1| < Rew} provided

|z f′(z)f(z)-1|≤2r1-r2 (∣c∣r2+2r+∣c∣r2+2∣c∣r+1+γr2+2r+γr2+2γr+1)≤12.

or equivalently, if r⁶+6(|c|+γ)r⁵+(17+20γ|c|)r⁴+24(|c|+γ)r³+(15+12γ|c|)r²+ 2(|c| +γ)r − 1 ≤ 0. Thus, the SP*

–radius of the class Fb2 is the smallest positive root r₃ ∈ (0, 1) of (3.13). The functions defined in (3.18) satisfy, for z = −r₃,

z f0′(z)f0(z)=1-21+r3-21-r3+2(1+cr3)1+2cr3+r32+2(1+2br3-cr3)1+r3(4b-2c+r3)=12,

which demonstrates sharpness.

Remark 3.6

Setting b = 1 = c, in Theorem 3.5, we obtain the result [4, Theorem 2.1] of Ali et al.

Let the functions f and g be given by (3.1) and (3.9) respectively. Assume that f and g are satisfying Re(f(z)/g(z)) > 0 and Re(g(z)/z) > 1/2 in . Then we have |a₂| ≤ |g₂| + 2 ≤ 3. In the following theorem we shall discuss some radius problems for functions with fixed second coefficients whose series expansion are given respectively by f(z) = z + 3bz² + · · · and g(z) = z + cz² + · · · with |b| ≤ 1 and |c| ≤ 1.

Definition 3.7

For |b| ≤ 1 and |c| ≤ 1, let

Fb,c3:={f∈A3b:Re (f(z)g(z))>0, and Re (g(z)z)>12, where g∈Ac}.

Now the radius constants for the class Fb,c3 are established in the following result.

Theorem 3.8

Assume thatγ₁ = |3b − c|. For the class Fb,c3,

the SL*–radius is the smallest positive root r₀ ∈ (0, 1) of
(3.19)2∣c∣r5+(1+2)(1+∣c∣γ1)r4+(6∣c∣+2(1+2)γ1)r3+(6+(3-2)∣c∣γ1)r2+2(2-1)(∣c∣+γ1)r-2+1=0
and it is sharp.
the ℳ(β)–radius is the smallest positive root r₁ ∈ (0, 1) of
(3.20)∣c∣βr5+(1+β)(1+∣c∣γ1)r4+(6∣c∣+(2+β)γ1)r3+(6+(3-β)∣c∣γ1)r2+(2-β)(∣c∣+γ1)r-β+1=0
and it is sharp.
the(α)–radius is the smallest positive root r₂ ∈ (0, 1) of
(3.21)-∣c∣αr7+(∣c∣(1-α)(γ1+2∣c∣)-1-α)r6+(∣c∣(2-α)(3+2∣c∣γ1)-αγ1)r5+(5+8∣c∣2-α+2(3-α)∣c∣γ1)r4+((12+α)∣c∣+2(2+∣c∣2α)γ1)r3+(5-2∣c∣2+α+2∣c2∣α+(1+3α)∣c∣γ1)r2+(2∣c∣+3∣c∣α+αγ1)r+α-1=0.
the SP*–radius is the smallest positive root r₃ ∈ (0, 1) of
(3.22)∣c∣r7+(1+4∣c∣2+3∣c∣γ1)r6+(17∣c∣+3γ1+8∣c∣2γ1)r5+(13+20∣c∣2+16∣c∣γ1)r4+(31∣c∣+8γ1+4∣c∣2γ1)r3+(11+5∣c∣γ1)r2+(γ1-∣c∣)r-1=0.

Proof

Define the functions p and h by p(z) = g(z)/z and h(z) = f(z)/g(z)

(3.23)p(z)=1+cz+⋯ and h(z)=f(z)g(z)=1+(3b-c)z+⋯

or p ∈ ℘_c_/2(1/2) and h ∈ ℘₍₃_b₋_c_)/2. Lemma 2.1 with α = 0 and α = 1/2 respectively lead to

(3.24)|zh′(z)h(z)|≤r1-r2γ1r2+4r+γ1r2+γ1r+1 and |zp′(z)p(z)|≤r1-r2∣c∣r2+2r+∣c∣∣c∣r+1.

From (3.23), f(z)/z = p(z)h(z), and so the inequalities in (3.24) yields

(3.25)|zf′(z)f(z)-1|≤|zh′(z)h(z)|+|zp′(z)p(z)|≤r1-r2 (γ1r2+4r+γ1r2+γ1r+1+∣c∣r2+2r+∣c∣∣c∣r+1).

(1) By Lemma 2.5, the function f satisfies |(zf′(z)/f(z))² − 1| < 1, for |z| < r, if the following inequality holds

r1-r2 (γ1r2+4r+γ1r2+γ1r+1+∣c∣r2+2r+∣c∣∣c∣r+1)≤2-1

or equivalently, if the following inequality holds:

2∣c∣r5+(1+2)(1+∣c∣γ1)r4+(6∣c∣+2(1+2)γ1)r3+(6+(3-2)∣c∣γ1)r2+2(2-1)(∣c∣+γ1)r-2+1≤0.

Therefore, the SL*-radius of the class Fb,c3 is the smallest positive root r₀ ∈ (0, 1) of (3.19). Consider the functions defined by

(3.26)f0(z)=z(1+(3b-c)z+z2)(1+cz)(1-z2)2 and g0(z)=z(1+cz)(1-z2).

The function f₀ with the choice g₀, defined in (3.26), is in the class Fb,c3 because

f0(z)g0(z)=1+w1(z)1-w1(z) and g0(z)z=1+w2(z)1-w2(z),

where w₁(z) = z(z + (3b − c)/2)/(1 + ((3b − c)z/2) with |3b − c| ≤ 2 and w₂(z) = z(z + c/2)/(1 + cz/2) are analytic functions satisfying the conditions of Schwarz’s lemma in the unit disk , and hence Re(g₀(z)/z) > 1/2 and Re(f₀(z)/g₀(z)) > 0 in . Since

z f0′(z)f0(z)=21-r0+21+r0-11+cr0-2+3br0-cr01+r0(3b-c+r0)=2,

for z = r₀, the root of (3.19), we have |(zf₀(z)/f₀(z))² −1| = 1. Thus, the result is sharp.

(2) The inequality (3.25) shows that

Re (z f′(z)f(z))≤r1-r2 (γ1r2+4r+γ1r2+γ1r+1+∣c∣r2+2r+∣c∣∣c∣r+1)+1≤β,

if the following inequality holds:

β∣c∣r5+(1+β)(1+∣c∣γ1)r4+(6∣c∣+(2+β)γ1)r3+(6+(3-β)∣c∣γ1)r2+(2-β)(∣c∣+γ1)r-β+1≤0.

Therefore the ℳ(β)–radius of the class Fb,c3 is the smallest positive root r₁ ∈ (0, 1) of (3.20). The result is sharp for the functions given in (3.26) as it can be seen that, for z = r₁, the root of (3.20), we have

z f0′(z)f0(z)=21-r1+21+r1-11+cr1-2+3br1-cr11+r1(3b-c+r1)=β.

(3) Since f(z)/z = p(z)h(z), it follows from Lemma 2.1 and Lemma 2.3 that

(3.27)Re (z f′(z)f(z))≥1-(γ1r2+4r+γ1)r(r2+γ1r+1)(1-r2)+(∣c∣+2r+∣c∣r2)r(1+2∣c∣r+r2)(1+∣c∣r)≥α,

if the following inequality holds:

-∣c∣αr7+(∣c∣(1-α)(γ1+2∣c∣)-1-α)r6+(∣c∣(2-α)(3+2∣c∣γ1)-αγ1)r5+(5+8∣c∣2-α+2(3-α)∣c∣γ1)r4+((12+α)∣c∣+2(2+∣c∣2α)γ1)r3+(5-2∣c∣2+α+2∣c∣2α+(1+3α)∣c∣γ1)r2+(2∣c∣+3∣c∣α+αγ1)r+α-1≤0.

Thus, the (α)–radius of the class Fb,c3 is the smallest positive root r₂ ∈ (0, 1) of (3.21).

(4) From (3.25) and (3.27), it is clear that |(zf′(z)/f(z))−1| < Re(zf′(z)/f(z)) provided

1-(γ1r2+4r+γ1)r(r2+γ1r+1)(1-r2)+(∣c∣+2r+∣c∣r2)r(1+2∣c∣r+r2)(1+∣c∣r)≥r1-r2 (γ1r2+4r+γ1r2+γ1r+1+∣c∣r2+2r+∣c∣∣c∣r+1)

or equivalently, if the following inequality holds:

∣c∣r7+(1+4∣c∣2+3∣c∣γ1)r6+(17∣c∣+3γ1+8∣c∣2γ1)+(13+20∣c∣2+16∣c∣γ1)r4+(31∣c∣+8γ1+4∣c∣2γ1)r3+(11+5∣c∣γ1)r2+(γ1+∣c∣)r-1≤0.

Thus the SP*–radius of the class Fb,c3 is the smallest positive root r₃ in (0, 1) of (3.22).

Remark 3.9

Putting b = 1 = c in Theorem 3.8, we obtain the result [4, Theorem 2.2] of Ali et al.

Now consider the functions f and g, given by (3.1) and (3.9) respectively. Suppose that f and g satisfy the conditions |f(z)/g(z) − 1| < 1 and Re(g(z)/z) > 0 in . Then it follows that |a₂| ≤ |g₂| + 2 ≤ 3. Thus such functions with fixed second coefficient, satisfying the above conditions can have the series expansion namely f(z) = z + 3bz² + · · · and g(z) = z + 2cz² + · · · with |b| ≤ 1 and |c| ≤ 1.

Definition 3.10

For |b| ≤ 1 and |c| ≤ 1, letFb,c4:={f∈A3b:|f(z)g(z)-1|<1,and Re (g(z)z)>0, where g∈A2c}.

Now in the following result, we provide the radius constants for the class Fb,c3.

Theorem 3.11

Assume thatδ := |2c − 3b|. For the class Fb,c4,

the SL*–radius is the smallest positive root r₀ ∈ (0, 1) of
(3.28)2δr5+(1+2)(1+2δ∣c∣))r4+2(3δ+2(1+2)∣c∣)r3+2(3+(3-2)δ∣c∣)r2+(2-2)(δ+2∣c∣)r-2+1=0.
the ℳ(β)–radius is the smallest positive root r₁ ∈ (0, 1) of
(3.29)βδr5+(1+β)(1+2δ∣c∣)r4+2(3δ+2∣c∣+∣c∣β)r3+2(3+(3-β)δ∣c∣)r2+(2-β)(δ+2∣c∣)r-β+1=0.
the f ∈ (α)–radius is the smallest positive root r₂ ∈ (0, 1) of
(3.30)(2-α)δr5+(1+2δ∣c∣)(3-α)r4+2(3δ+4∣c∣-∣c∣α)r3+(δ+2∣c∣)αr2+2(3+(1+α)δ∣c∣)r+α-1=0.
the SP*–radius is the smallest positive root r₃ ∈ (0, 1) of
(3.31)3δr5+5(1+2δ∣c∣)r4+2(6δ+7∣c∣)r3+6(2+δ∣c∣)r2+(δ+2∣c∣)r-1=0.

Proof

It is easy to see that |f(z)/g(z) − 1| < 1 if and only if Re(g(z)/f(z)) > 1/2. Define the functions p and h by p(z) = g(z)/z, and h(z) = g(z)/f(z). Then

p(z)=1+2cz+⋯ and h(z)=g(z)f(z)=1+(2c-3b)z+⋯

or p ∈ ℘_c and h ∈ ℘₍₂_c₋₃_b_)/2(1/2). Lemma 2.1 with α = 0 and α = 1/2 respectively lead to

(3.32)|zh′(z)h(z)|≤2r(∣c∣r2+2r+∣c∣)(1-r2)(r2+2∣c∣r+1) and |zp′(z)p(z)|≤r(δr2+2r+δ)(1-r2)(δr+1)

respectively, where δ := |2c − 3b|. Since zp(z) = f(z)h(z), from (3.32), we have

(3.33)|zf′(z)f(z)-1|≤|zp′(z)p(z)|+|zh′(z)h(z)|≤r1-r2 (2(∣c∣r2+2r+∣c∣)r2+2qr+1+(δr2+2r+δ)δr+1).

(1) By Lemma 2.5, the function f satisfies |(zf′(z)/f(z))² − 1| < 1, if the following inequality holds:

r1-r2 (2(∣c∣r2+2r+∣c∣)r2+2qr+1+(δr2+2r+δ)δr+1)≤2-1,

or equivalently, if

2δr5+(1+2)(1+2δ∣c∣))r4+2(3δ+2(1+2)∣c∣)r3+2(3+(3-2)δ∣c∣)r2+(2-2)(δ+2∣c∣)r-2+1≤0.

Therefore the SL*–radius of the class Fb,c4 is the smallest positive root r₀ ∈ (0, 1) of (3.28).

(2) Using (3.33), we get

Re (zf′(z)f(z))≤1+r1-r2 (2(∣c∣r2+2r+∣c∣)r2+2qr+1+(δr2+2r+δ)δr+1)≤β

if the following inequality holds:

βδr5+(1+β)(1+2δ∣c∣)r4+2(3δ+2∣c∣+∣c∣β)r3+2(3+(3-β)δ∣c∣)r2+(2-β)(δ+2∣c∣)r-β+1≤0.

Therefore, the ℳ(β)–radius of the class Fb,c4 is the smallest positive root r₁ ∈ (0, 1) of (3.29).

(3) Inequality in (3.33) implies that

Re (zf′(z)f(z))≥1-r1-r2 (2(∣c∣r2+2r+∣c∣)r2+2qr+1+(δr2+2r+δ)δr+1)≥α

if the following inequality holds:

(2-α)δr5+(1+2δ∣c∣)(3-α)r4+2(3δ+4∣c∣-∣c∣α)r3+(δ+2∣c∣)αr2+2(3+(1+α)δ∣c∣)r+α-1≤0.

Thus, the (α)–radius of the class Fb,c4 is the smallest positive root in r₂ ∈ (0, 1) of (3.30).

(4) Lemma 2.6 shows that the disk (3.33) lies inside Ω = {w : |w − 1| < Rew}, the parabolic region, provided

|zf′(z)f(z)-1|≤r1-r2 (2(∣c∣r2+2r+∣c∣)r2+2qr+1+(δr2+2r+δ)δr+1)≤12

if the following inequality holds:

3δr5+5(1+2δ∣c∣)r4+2(6δ+7∣c∣)r3+6(2+δ∣c∣)r2+(δ+2∣c∣)r-1≤0.

Therefore, the SP*–radius of the class Fb,c4 is the smallest positive root r₃ ∈ (0, 1) of (3.31).

Remark 3.12

Note that, in addition, we obtain the following sharp results of [4, Theorem 2.3] of Ali et al. as special case to parts (3) and (4) of Theorem 3.11 when b = 1 = c.

For the class F1,14,

the SL*–radius,r0=2(2-2)2(17-42+3)
the ℳ(β)–radius, r1=2(β-1)3+9+4β(β-1),
the sharp f ∈ (α)–radius, r2=2(1-α)3+9+4β(1-α)(2-α),
the sharp SP*–radius, r3=23-33.

Consider the functions f and g given by (3.1) and (3.9) respectively. Further assume that f and g satisfy the condition |f(z)/g(z) − 1| < 1 and g is a convex function in the unit disk . Then we have |a₂| ≤ |g₂|+1 ≤ 2. In the next theorem, we consider such functions with fixed second coefficient, whose series expansion are given by f(z) = z + 2bz² + · · · and g(z) = z + cz² + · · · with |b| ≤ 1 and |c| ≤ 1.

Definition 3.13

For |b| ≤ 1 and |c| ≤ 1, let

Fb,c5:={f∈A2b:Re (f(z)g(z))>0, where g∈Ac∩K=Kc}.

We now obtain the radius constants for the class Fb,c3 in the following result.

Theorem 3.14

Assume thatδ₁ := |c − 2b|. For the class Fb,c5,

the(λ)–radius is the smallest root r₀ ∈ (0, 1) of
(3.34)(δ1+β0δ1-δ1λ)r5+(2+β0+3∣c∣δ1+∣c∣β0δ1-λ-2∣c∣δ1λ)r4+(5∣c∣+∣c∣β0+3δ1-β0δ1-2∣c∣λ)r3+(3-β0+(1-β0+2λ)δ1∣c∣)r2+(2∣c∣λ+δ1λ-∣c∣-∣c∣β0)r+λ-1=0,
whereβ₀ = 2α₀−1 andα₀ ∈ (0, 1) is the smallest positive root of the equation
2α3-qα2-4α+2=0
in the interval [1/2, 2/3].
the SP*–radius is the smallest root r₁ ∈ (0, 1) of
(3.35)(δ1+2β0δ1)r5+(3+2β0+4∣c∣δ1-2∣c∣β0δ1)r4+(8∣c∣-2∣c∣β0+6δ1-2β0δ1+2∣c∣β0δ1-2∣c∣2β0δ1)r3+(6-2β0+2∣c∣β0-2∣c∣2β0+4∣c∣δ1-2∣c∣β0δ1)r2+(-2∣c∣β0+δ1)r-1=0.
the SL*–radius is the smallest root r₂ ∈ (0, 1) of
(3.36)(δ1+2δ1-β0δ1)r5+(2+2-β0+3qδ1+22∣c∣δ1-∣c∣β0δ1)r4+(5∣c∣+22∣c∣-∣c∣β0+3δ1+2∣c∣2δ1-β0δ1-∣c∣β0δ1-∣c∣2β0δ1)r3+(3+2∣c∣2-β0-∣c∣β0-∣c∣2β0+5∣c∣δ1-22∣c∣δ1-∣c∣β0δ1)r2+(3∣c∣-22∣c∣-∣c∣β0+2δ1-2δ1)r-2+1=0.
the ℳ(β)–radius is the smallest root r₃ ∈ (0, 1) of
(3.37)(δ1+βδ1-β0δ1)r5+(2+β-β0+3∣c∣δ1+2β∣c∣δ1-∣c∣β0δ1)r4+(5∣c∣+2β∣c∣-∣c∣β0+3δ1+2∣c∣2δ1-β0δ1-∣c∣β0δ1-∣c∣2β0δ1)r3+(3+2∣c∣2-β0-∣c∣β0-∣c∣2β0+5∣c∣δ1-2b∣c∣δ1-∣c∣β0δ1)r2+(∣c∣-2β∣c∣-∣c∣β0+2δ1-βδ1)r-β+1=0.

Proof

Define the functions h and p by h(z) = g(z)/f(z), and p(z) = zg′(z)/g(z). Then

h(z)=1+(c-2b)z+⋯ and p(z)=1+cz+⋯.

Since |f(z)/g(z) − 1| < 1 if and only if Re(g(z)/f(z)) > 1/2, we have h ∈ ℘₍_c₋₂_b_)/2(1/2). An application of Lemma 2.1 to the function h(z), gives

(3.38)|zh′(z)h(z)|≤(δ1r2+2r+δ1)r(δ1r+1)(1-r2),

where δ₁ := |c − 2b|. Since g(z) = z + cz² + · · · ∈ , it follows from Lemma that

Re (zg′(z)g(z))>α0,

where α₀ is the smallest positive root of the equation 2α³ − |c|α² − 4α + 2 = 0 in the interval [1/2, 2/3]. Thus Re(p(z)) > α₀.

(1) An application of Lemma with α = α₀, gives

(3.39)∣p(z)-Cc∣≤Dc,

where

Cc=(1+∣c∣r)2-β0(∣c∣+r)2r2(1+2∣c∣r+r2)(1-r2), Dc=(1-β0)(∣c∣+r)(1+∣c∣r)r(1+2∣c∣r+r2)(1-r2) and β0=2α0-1.

and β₀ = 2α₀−1.

Since h(z) = g(z)/f(z) and p(z) = zg′(z)/g(z), we have

(3.40)|zf′(z)f(z)-Cc|≤∣p(z)-Cc∣+|zh′(z)h(z)|.

From (3.39), (3.38) and (3.40), we have

(3.41)|zf′(z)f(z)-Cc|≤Dc+(δ1r2+2r+δ1)r(δ1r+1)(1-r2).

Clearly f ∈ (λ), provided that

Re (zf′(z)f(z))≥Cc-Dc-(δ1r2+2r+δ1)r(δ1r+1)(1-r2)≥λ

or equivalently, if the following inequality holds:

(δ1+β0δ1-δ1λ)r5+(2+β0+3∣c∣δ1+∣c∣β0δ1-λ-2∣c∣δ1λ)r4+(5∣c∣+∣c∣β0+3δ1-β0δ1-2∣c∣λ)r3+(3-β0+(δ1-β0δ1+2δ1λ)∣c∣)r2+(-∣c∣-∣c∣β0+2∣c∣λ+δ1λ)r-1+λ≤0.

Thus, the (λ)–radius of the class Fb,c5 is the smallest positive root r₀ ∈ (0, 1) of (3.34).

(2) In view of Lemma 2.6, the disk given in (3.41) lies inside the parabolic region given by Ω := {w : |w − 1| < Rew} provided

Dc+(δ1r2+2r+δ1)r(δ1r+1)(1-r2)≤Cc-1/2

or equivalently, if the following inequality holds:

(δ1+2β0δ1)r5+(3+2β0+4∣c∣δ1-2∣c∣β0δ1)r4+(8∣c∣-2∣c∣β0+6δ1-2β0δ1+2∣c∣β0δ1-2∣c∣2β0δ1)r3+(6-2β0+2∣c∣β0-2∣c∣2β0+4∣c∣δ1-2∣c∣β0δ1)r2+(δ1-2∣c∣β0)r-1≤0.

Hence the ℳ(β)–radius of the class Fb,c5 is the smallest positive root r₁ ∈ (0, 1) of (3.35).

(3) From Lemma 2.5, the function f satisfies |(zf′(z)/f(z))² − 1| < 1, in |z| < r, if the following inequality holds:

Dc+(δ1r2+2r+δ1)r(δ1r+1)(1-r2)≤2-Cc,

or equivalently, if the following inequality holds:

(δ1+2δ1-β0δ1)r5+(2+2-β0+3qδ1+22∣c∣δ1-∣c∣β0δ1)r4+(5∣c∣+22∣c∣-∣c∣β0+3δ1+2∣c∣2δ1-β0δ1-∣c∣β0δ1-∣c∣2β0δ1)r3+(3+2∣c∣2-β0-∣c∣β0-∣c∣2β0+5∣c∣δ1-22∣c∣δ1-∣c∣β0δ1)r2+(3∣c∣-22∣c∣-∣c∣β0+2δ1-2δ1)r-2+1≤0.

Therefore the SL*–radius of the class Fb,c5 is the smallest positive root r₂ ∈ (0, 1) of (3.36).

(4) From (3.41), we have

Re (zf′(z)f(z))≤Cc+Dc+(δ1r2+2r+δ1)r(δ1r+1)(1-r2)≤β.

if the following inequality holds:

(δ1+βδ1-β0δ1)r5+(2+β-β0+3∣c∣δ1+2β∣c∣δ1-∣c∣β0δ1)r4+(5∣c∣+2β∣c∣-∣c∣β0+3δ1+2∣c∣2δ1-β0δ1-∣c∣β0δ1-∣c∣2β0δ1)r3+(3+2∣c∣2-β0-∣c∣β0-∣c∣2β0+5∣c∣δ1-2b∣c∣δ1-∣c∣β0δ1)r2+(∣c∣-2β∣c∣-∣c∣β0+2δ1-βδ1)r-β+1=0.

Therefore, the ℳ(β)–radius of the class Fb,c5 is the smallest positive root r₃ ∈ (0, 1) of (3.37).

Remark 3.15

Note that for b = 1 = c, Theorem 3.14 reduces to the result [4, Theorem 2.5] of Ali et al.

Acknowledgements

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Abstract
1. Introduction
2. Preliminaries
3. Radius constants
Acknowledgements
References

The work presented here was supported in parts by a research university grant from Universiti Sains Malaysia, and a SRF from Delhi Technological University.

References

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Abstract
1. Introduction
2. Preliminaries
3. Radius constants
Acknowledgements
References

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