Certain Estimates of Normalized Analytic Functions

Swati Anand; Naveen Kumar Jain; Sushil Kumar

doi:10.1515/ms-2022-0006

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Certain Estimates of Normalized Analytic Functions

Swati Anand , Naveen Kumar Jain and Sushil Kumar

Published/Copyright: February 16, 2022

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From the journal Mathematica Slovaca Volume 72 Issue 1

Abstract

Let ϕ be a normalized convex function defined on open unit disk D. For a unified class of normalized analytic functions which satisfy the second order differential subordination f′(z) + αzf″(z) ≺ ϕ(z) for all z∈D, we investigate the distortion theorem and growth theorem. Further, the bounds on initial logarithmic coefficients, inverse coefficient and the second Hankel determinant involving the inverse coefficients are examined.

MSC 2010: Primary 30C45; 30C50; 30C80

Keywords: Differential subordination; growth theorem; distortion theorem; logarithmic coefficient; inverse coefficient; Hankel determinant

1. Introduction

The class of all normalized analytic functions

(1.1)fz=z+a2z2+a3z3+⋯

in the unit disk D:={z∈ℂ:|z|<1} is denoted by A. Denote by S, the subclass of A consisting of univalent functions in D. Let P be the class of analytic functions p defined on D, normalized by the condition p(0) = 1 and satisfying Re(p(z)) > 0. Let f and g be analytic in D. Then f is subordinate to g, denoted by f ≺ g, if there exists an analytic function w with w(0) = 0 and |w(z)| < 1 for z∈D such that f (z) = g(w(z)). In particular, if g is univalent in D then f is subordinate to g when f(0) = g(0) and f(D)⊆g(D). In this paper we shall assume that ϕ is an analytic function with positive real part in D and normalized by the conditions ϕ(0) = 1 and ϕ′(0) > 0. It is noted that ϕ(D) is convex. The function ϕ is symmetric with respect to the real axis and it maps D onto a region starlike with respect to ϕ(0) = 1. The Taylor series representation of the function ϕ is given by

(1.2)ϕz=1+B1z+B2z2+B3z3+⋯,

where B₁ > 0. For such ϕ, Ma and Minda [22] studied the unified subclasses S*(ϕ) and C(ϕ) of starlike and convex functions respectively, analytically defined as

S*ϕ=f∈A:zf′zfz≺ϕzandCϕ=f∈A:1+zf″zf′z≺ϕz.

The authors investigated the growth, distortion and coefficient estimates for these classes. In particular, the class S*(ϕ) reduces to some well known subclasses of starlike fuctions. For example, when −1 ≤ B < A ≤ 1, S*[A, B] is the class of Janowski starlike functions introduced by Janowski [11]. For 0 ≤ α < 1, S*[1−2α,−1]=Sα* is the class of starlike functions of order α, introduced by Robertson [31]. The class Sℒ=S*(1+z), introduced by Sokół and Stankiewicz [36], consists of functions f∈A such that zf′(z)/f(z) lies in the region ΩL := {w ∈ ℂ:|w2−1|<1}, that is, the right-half of the lemniscate of Bernoulli. Later, Mendiratta et al. [23] introduced the class S*e:=S*(ez) consists of functions f∈A satisfying the condition |log(zf′(z)/f′(z))| < 1. In 2011, Ali et al. [3](see also [10]) studied the class of all those functions f∈A which satisfy the following third order differential equation

fz+αf′z+γz2f″z=gz,

where the function g is subordinate to a convex function h. In [3], the best dominant on all solutions of the differential equation in terms of double integral were obtained. Some certain variations of the class ℛ(α,h)={f∈A:f′(z)+αzf″(z)≺h(z),z∈D}, where h is a convex function have been investigated by several authors [6,24,35,37,39]. On the basis of the above discussed works, we consider a unified class of all functions f∈A such that

(1.3)f′z+αzf″z≺ϕz

for z ∈ D and where α ∈ ℂ with Re α ≥ 0. The class of such functions is denoted by ℛ(α,ϕ). Since f ∈ R(α, ϕ), f′(z) + αzf″(z) ≠ ϕ(e^iθ), θ ∈ [0, 2π), it is observed that

f′(z)+αzf″(z)=[(1−α)f(z)+α(zf′(z))]′.

Also, we have zf′(z)=f(z)*z(1−z)2 and f(z)=f(z)*z1−z. Thus,

f′z+αzf″z=1−αfz∗z1−z+αfz∗z1−z2′=fz∗1−αz1−z+αz1−z2′.

Therefore, we conclude that

fz∗z−z21−α1−z2′−ϕeiθ≠0

or equivalently

1zfz∗z+z22α−11−z3≠ϕeiθ

which is the necessary and sufficient conditions for a function f∈A to be in the class ℛ(α,ϕ).

In this paper, we compute the distortion, growth inequalities for a function f in the class ℛ(α,ϕ). The sharp bounds on initial logarithmic coefficients for such functions are also obtained. Next, we obtain the bounds on initial inverse coefficients of the function f∈ℛ(α,ϕ) as well as bounds on Fekete Szegö functional and second Hankel determinant.

2. Distortion and growth theorem

The first theorem proves the distortion theorem of the functions f belonging to ℛ(α,ϕ).

THEOREM 2.1

Letα ∈ ℂ, Re α ≥ 0 and the functionϕ be defined by (1.2). If the functionf∈ℛ(α,ϕ), then

1+∑n=1∞|Bn|−rnnReα+1≤|f′z|≤1+∑n=1∞|Bn|rnnReα+1

for |z| < r < 1. The result is sharp.

We make use of the following lemma in order to prove some of our results.

LEMMA 2.1

([7: Lemma 2, p. 192]). Let h be a convex function with γ ≠ 0 and Re γ ≥ 0. Ifp(z) is regular inDandp(0) = h(0), then

(2.1)pz+zp′zγ≺hz

implies thatp(z) ≺ q(z) ≺ h(z), where

qz=γz−γ∫0zhttγ−1dt.

The function q is convex and the best dominant.

Proof of Theorem 2.1. Let the fuction f be in the class ℛ(α,ϕ). Then f′(z) + αzf′(z) ≺ ϕ(z). For p(z) = f′(z) and γ = 1/α, Lemma 2.1 yields

f′z≺1αz−1/α∫0zϕtt1/α−1dt.

On taking t = zζ^α in above relation, we get

(2.2)f′z≺∫01ϕzζαdζ.

Since the function ϕ is symmetric with respect to real axis, ϕ(z) has real coefficients. Also ϕ′(0) > 0 gives ϕ′(x) is increasing on (0, 1). From [12], we have

(2.3)min |z|=rRe ϕ(z)=ϕ(−r)andmax|z|=rReϕ(z)=ϕ(r).

Using (2.3) and (2.2) for |z| = r, we have

(2.4)|f′z|≥Ref′z≥min|z|=rRef′z≥min|z|=rRe∫01ϕzζαdζ=∫01min|z|=rReϕzζαdζ=∫01ϕ−rζReαdζ.

Similarly, we have

(2.5)|f′z|≤∫01ϕrζReαdζ.

Since ϕ(zζ^{Re α}) = 1 + B₁zζ^{Re α} + B₂z²ζ^{2Re α} + ···, a simple calculation yields

(2.6)∫01ϕzζReαdζ=∫011+B1zζReα+B2z2ζ2Reα+B3z3ζ3Reα+⋯dζ=1+B1zReα+1+B2z22Reα+1+B3z33Reα+1+⋯=1+∑n=1∞BnznnReα+1.

From (2.4), (2.5) and (2.6) the result follows. The result is sharp for the function f:D→ℂ defined by

(2.7)fz=z+∑n=1∞Bnzn+1n+11+nReα.

THEOREM 2.2.

Letα∈ℂsuch that Re α ≥ 0 and the function ϕ be as in (1.2). Then for the function f ∈ R(α, ϕ), we have

1+∑n=1∞|Bn|−rnn+1nReα+1≤|fz||z|≤1+∑n=1∞|Bn|rnn+1nReα+1|z|<r<1.

Proof. Let

(2.8)Hz=∫01ϕzζαdζ

and

Φαz=∫0111−ztαdt=∑n=0∞zn1+nα.

From [33: Theorem 5, p. 113], it is noted that Φ_α(z) is convex with Re α ≥ 0. Also,

Φαz∗ϕz=∑n=0∞zn1+nα∗1+∑n=1∞Bnzn=1+∑n=1∞Bn1+nαzn.

It view of above and (2.6), we have

Φαz∗ϕz=∫01ϕzζαdζ=Hz.

Since convolution of two convex functions is convex, the function H is convex and H(0) = 1. Putting γ = 1 and h(z) = H(z) in Lemma 2.1, we get

(2.9)pz≺1z∫0zHtdt.

Using (2.8), substituting t = zσ and p(z) = f(z)/z in (2.9), we have

(2.10)fzz≺∫01∫01ϕzσζαdσdζ.

Let h(z) = f(z)/z. Then (2.3) together with (2.10) yields

(2.11)|hz|≥min|z|=r Rehz≥∫01∫01min|z|=rϕzσζαdσdζ=∫01∫01ϕ−rσζRe αdσdζ

and

(2.12)|hz|≤max|z|=r Re hz≤∫01∫01max|z|=rϕzσζαdσdζ=∫01∫01ϕ−rσζRe αdσdζ.

A simple calculation shows that

(2.13)∫01∫01ϕzσζRe αdσdσζ=∫011+∑n=1∞Bn1+nRe αzσndσ=1+∑n=1∞Bnznn+11+nRe α.

Now, the result follows from (2.11), (2.12) and (2.13). The result is sharp for the function f given by (2.7).

Remark 1.

Letting ϕ(z) = (1 – (1 – 2β)z)/(1 – z), where β < 1 and α = 1, Theorem 2.2 reduces to a result due to Silverman [34: Corollary 2, p. 250]. Further, for ϕ(z) = (1 – (1 – 2β)z)/(1 – z), where β < 1 and α > 0, Theorem 2.3 yields [6: Corollary 3, p. 178].

THEOREM 2.3.

Letα∈ℂsuch that Re α ≥ 0 and the function ϕ be as in (1.2) such thatf ∈ R(α, ϕ). Then

(2.14)|an| ≤ B1|n+nn−1α|foralln≥2.

Proof. For p(z)=1+∑n=1∞pnzn∈P, set f′(z) + αzf″(z) = p(z), z∈D. Since f ∈ R(α, ϕ), p(z) ≺ ϕ(z). A simple calculation gives

(2.15)f′z+αzf″z=1+∑n=2∞[n+nn−1α]anzn−1=1+∑n=1∞pnzn.

On comparing the coefficients of z^n–1, we get

n+nn−1αan=pn−1foralln≥2.

By making use of [32: Theorem X, p. 70], we get |p_n| ≤ B₁, for all n ≥ 1. Hence, we get the desired inequality. The inequality (2.14) is sharp for the function f_n given by fn′(z)+αzfn″(z)=ϕ(zn−1).

Remark 2.

On taking ϕ(z) = (1 – (1 – 2β)z)/(1 – z), where β < 1 and α =1, Theorem 2.3 yields a result due to Silverman [34: Corollary 1, p. 250]. Further, letting ϕ(z) = (1 – (1 – 2β)z)/(1 – z), where β < 1 and α > 0 Theorem 2.3 reduces to [6: Corollary 2, p. 178].

3. Bounds on initial logarithmic coefficient

For a function f∈S, the logarithmic coefficients γ_n are defined by the following series expansion:

(3.1)logfzz=2∑n=1∞γnzn,log1:=0.

On comparing the coefficients of z on both the sides, we get the initial logarithmic coefficients

(3.2)γ1=12a2,γ2=12a3−12a22γ3=12a4−a2a3+13a23

In 1979, the authors [5] showed that the logarithmic coefficients γ_n of every function f∈S satisfy the inequality ∑n=1∞|γn|2≤π2/6, where the equality holds if and only if the function f is rotation of the Koebe function k(z) = z(1 – e^iθ)⁻² for each θ. The n^th logarithmic coefficient of k(z) is γ_n = e^inθ/n for each θ and n ≥ 1. In [40], the logarithmic coefficients γ_n of each close-to-convex function f ∈ S is bounded by (A log n)/n where A is an absolute constant. In 2018, the authors [4,28] obtained the bounds on logarithmic coefficients of certain subclasses of the class of close-to-convex functions. Recently, Adegani et al. [1] investigated the bounds for the initial logarithmic coefficients of the generalized classes S*(ϕ) and C(ϕ). To find the bounds on initial logarithmic coefficient for class ℛ(α,ϕ), we shall use the following two lemmas.

LEMMA 3.1

([26: p. 172]). Assume that w is a Schwarz function so thatw(z)=∑n=1∞cnzn. Then

|c1|≤1and|cn|≤1−|c1|2,n=2,3,….

Lemma 3.2

([29: Theorem 1]). Letw(z)=∑n=1∞cnznbe the Schwarz function. Then for any real numbers q₁ and q₂, the following sharp inequality holds:

|c3+q1c1c2+q2c13|≤Hq1;q2,

where

Hq1;q2=1ifq1,q2∈D1∪D2∪2,1,|q2|ifq1,q2∈∪k=17Dk,23|q1|+11+|q1|3|q1|+1+q212ifq1,q2∈D8∪D9,q23q12−4q12−4q2q12−43q2−112ifq1,q2∈D10∪D11\2,1,23|q1|−1|q1|−13|q1|−1−q212ifq1,q2∈D12,

and for k = 1, 2, …, 12, the sets D_k are defined as follows:

D1=q1,q2:q1≤12,q2≤1,D2=q1,q2:12≤q1≤2,427q1+13−q1+1≤q2≤1,D3=q1,q2:q1≤12,q2≤−1,D4=q1,q2:q1≥12,q2≤−23q1+1,D5=q1,q2:q1≤2,q2≥1,D6=q1,q2:2≤q1≤4,q2≥112q12+8,D7=q1,q2:q1≥4,q2≥23q1−1,D8=q1,q2:12≤q1≤2,−23q1+1≤q2≤427q1+13−q1+1,D9=q1,q2:q1≥2,−23q1+1≤q2≤2q1q1+1q12+2q1+4,D10=q1,q2:2≤q1≤4,2q1q1+1q12+2q1+4≤q2≤112q12+8,D11=q1,q2:q1≥4,2q1q1+1q12+2q1+4≤q2≤2q1q1−1q12−2q1+4,D12=q1,q2:q1≥4,2q1q1−1q12−2q1+4≤q2≤23q1−1.

THEOREM 3.1

Letα∈ℂsuch that Re α ≥ 0 and the function ϕ be as in (1.2). Supposef∈ℛ(α,ϕ), then the initial logarithmic coefficients of f satisfy the following inequalities:

|γ1|≤B14|1+α|.
|γ2|≤{B16|1+2α|if|8(1+α)2B2−3(1+2α)B12|≤8B1|(1+α)2|,|8(1+α)2B2−3(1+2α)B12|48|(1+α)2||1+2α|if|8(1+α)2B2−3(1+2α)B12|>8B1|(1+α)2|.
If B₁, B₂, B₃and a are real numbers, then
|γ3|≤B18(1+3α)H(q1;q2),
where H(q₁; q₂) is given inLemma 3.2such that
q1=2B2B1−2B1(1+3α)3(1+α)(1+2α)
and
q2=B3B1−2B2(1+3α)3(1+α)(1+2α)+B12(1+3α)6(1+α)3.

The bounds for γ₁and γ₂are sharp.

Proof. Let f(z)=z+∑n=2∞anzn∈ℛ(α,ϕ) where ϕ is given by (1.2). Then f′(z) + αzf″(z) = ϕ(w(z)) for z∈D, where w(z)=∑n=1∞cnzn is the Schwarz function. A simple calculation yields

f′(z)+αzf″(z)=1+B1c1z+(B1c2+B2c12)z2+(B1c3+2c1c2B2+B3c13)z3+⋯.

On comparing the coefficients of z, we obtain

2(1+α)a2=B1c1,3(1+2α)a3=B1c2+B2c12,4(1+3α)a4=B1c3+2B2c1c2+B3c13.

On substituting the above values of a_i(i = 1, 2, 3) in (3.2), we get

(3.3)γ1=B1c14(1+α),γ2=8(1+α)2B1c2+(8(1+α)2B2−3(1+2α)B12)c1248(1+α)2(1+2α),γ3=B18(1+3α)c3+(B24(1+3α)+B1212(1+α)(1+2α))c1c2+(B38(1+3α)−B1B212(1+α)(1+2α)+B1348(1+α)3)c13.

By using Lemma 3.1, we get the desired best possible estimate on γ₁. The bound is sharp for |c₁| = 1.

|γ2|=|B16(1+2α)c2+8(1+α)2B2−3(1+2α)B1248(1+α)2(1+2α)c12|≤B16|1+2α||c2|+|8(1+α)2B2−3(1+2α)B12|48|(1+α)2||1+2α||c12|≤B16|1+2α|(1−|c1|2)+|8(1+α)2B2−3(1+2α)B12|48|(1+α)2||1+2α||c12|=B16|1+2α|+(|8(1+α)2B2−3(1+2α)B12|48|(1+α)2||1+2α|−B16|1+2α|)|c12|≤{B16|1+2α|if|8(1+α)2B2−3(1+2α)B12|48|(1+α)2||1+2α|≤B16|1+2α||8(1+α)2B2−3(1+2α)B12|48|(1+α)2||1+2α|if|8(1+α)2B2−3(1+2α)B12|48|(1+α)2||1+2α|>B16|1+2α|.

These bounds are sharp for |c₁| = 0 and |c₁| = 1, respectively.

The third inequality follows by Lemma 3.1. Using Lemma 3.2 for γ₃, we obtain

|γ3|=|B18(1+3α)c3+(B24(1+3α)−B1212(1+α)(1+2α))c1c2+(B38(1+3α)−B1B212(1+α)(1+2α)+B148(1+α)3)c13|=B18(1+3α)|c3+c1c2q1+c13q2|≤B18(1+3α)H(q1;q2),

where

q1=2(B2B1−B1(1+3α)3(1+α)(1+2α))andq2=B3B1−2B2(1+3α)3(1+α)(1+2α)+B12(1+3α)6(1+α)3.

This completes the proof. □

On taking ϕ(z) = e^z, ϕ(z)=1+z and ϕ(z) = (1 + z)/(1 – z), respectively in Theorem 3.1 the following corollaries follows immediately.

Corollary 3.1.1

Letα∈ℂsuch that Re α ≥ 0 and ϕ(z) = e^z. Supposef∈ℛ(α,ϕ), then the initial logarithmic coefficients of f satisfy the following inequalities:

|γ1|≤14|1+α|,
|γ2|≤16|1+2α|,
|γ3|≤18(1+3α)H(q1;q2),

where H(q₁; q₂) is given inLemma 3.2such that

q1=1+3α(1+2α)3(1+α)(1+2α)

and

q2=16−(1+3α)3(1+α)(1+2α)+(1+3α)6(1+α)3.

Corollary 3.1.2.

Suppose thatf∈ℛ(α,ϕ)where Re α ≥ 0 andϕ(z)=1+z, then the initial logarithmic coefficients of f satisfy the following inequalities:

|γ1|≤18|1+α|,
|γ2|≤112|1+2α|,
|γ3|≤116(1+3α)H(q1;q2),

where H(q₁; q₂) is given inLemma 3.2such that

q1=−12−(1+3α)3(1+α)(1+2α)

and

q2=18+1+3α12(1+α)(1+2α)+1+3α24(1+α)3.

Corollary 3.1.3.

Let the functionf∈ℛ(α,ϕ)where Re α ≥ 0 and ϕ(z) = (1 + z)/(1 – z), then the initial logarithmic coefficients of f satisfy the following inequalities:

|γ1|≤12|1+α|,
|γ2|≤13|1+2α|,
|γ3|≤14(1+3α)H(q1;q2),

where H(q₁; q₂) is given inLemma 3.2such that

q1=2(1+3α(1+2α))3(1+α)(1+2α)

and

q2=1−4(1+3α)3(1+α)(1+2α)+2(1+3α)3(1+α)3.

4. Inverse coefficient estimates

From the Koebe one quarter theorem, the image of D under a function f∈S contains a disk of radius 1/4. Thus for every univalent function f there exist inverse function f⁻¹ such that f⁻¹(f(z)) = z for z∈D and f(f⁻¹(ω)) = ω for |ω| < r₀(f) where r₀(f) ≥ 1/4. The function f⁻¹ has the Taylor series expansion f⁻¹(ω) = ω + A₂ω² + A₃ω³ + … in some neighborhood of origin. In 1923, Löwner [21] initiated the problem of estimating the coefficients of inverse function and investigated the coefficient estimates for inverse function f∈S. Later on, this lead to the study of inverse coefficient problem for several subclasses of S by various authors [2, 18–20, 29]. In [13, 16], authors obtained the initial inverse coefficients for the well known classes C and S*(α) (0 ≤ α ≤ 1). Recently, Ravichandran and Verma [30] determined the bounds on inverse coefficient for the Janowski starlike functions.

In this section, we shall investigate the bounds on inverse coefficient. The following lemma is needed to obtain the coefficient bounds for the inverse function.

Lemma 4.1

([18: Lemma 3, p. 254]). If p(z) = 1 + p₁z + p₂z² + … is a function in the class P, then for any complex number ν,

|p2−vp12|≤2max{1,|2v−1|}.

THEOREM 4.1

Letα∈ℂsuch that Re α ≥ 0 and the function ϕ defined by (1.2). If functionf(z)=z+∑n=2∞anzn∈ℛ(α,ϕ)andf−1(ω)=ω+∑n=2∞Anωnfor all ω in some neighbourhood of the origin, then

|A2|≤B12|1+α|,
|A3|≤B13|1+2α|max{1,|μ|}, whereμ=3B1(1+2α)2(1+α)2−B2B1,
If B₁, B₂, B₃and α are real numbers, then
|A4|≤B14(1+3α)H(q1;q2),

where H(q₁; q₂) is given inLemma 3.2such that

q1=2(B2B1−5B1(1+3α)3(1+α)(1+2α))

and

q2=B3B1−10B2(1+3α)3(1+α)(1+2α)+5B12(1+3α)2(1+α)3.

Proof. Let f∈ℛ(α,ϕ). Then

(4.1)f′(z)+αzf″(z)=ϕ(w(z)),

where w(z) is the analytic function w with w(0) = 0 and |w(z)| < 1. Let

p(z)=1+w(z)1−w(z)=1+p1z+p2z2+p3z3+⋯.

Since w:D→D is analytic thus p is a function with positive real part and

(4.2)w(z)=p(z)−1p(z)+1=12p1z+12(p2−p122)z2+18(p13−4p1p2+4p3)z3+⋯.

Then

(4.3)ϕ(w(z))=1+B1p12z+(14B2p12+12B1(p2−12p12))z2+18((B1−2B2+B3)p13+4(−B1+B2)p1p2+4B1p3)z3+⋯.

Using expressions (4.3) and (4.1), we obtain

(4.4)2(1+α)a2=B1p12,3(1+2α)a3=14B2p12+12B1(p2−12p12),and4(1+3α)a4=18((B1−2B2+B3)p13+4(−B1+B2)p1p2+4B1p3).

Since f⁻¹(ω) = ω + A₂ω² + A₃ω³ + A₄ω⁴ + ··· in some neighbourhood of origin, so we have f(f⁻¹(ω)) = ω. That is

ω=f−1(ω)+a2(f−1(ω))2+a3(f−1(ω))3+⋯=ω+A2ω2+A3ω3+A4ω4+⋯+a2(ω+A2ω2+A3ω3+A4ω4+⋯)2+a3(ω+A2ω2+A3ω3+A4ω4+⋯)3.

A simple calculation gives the following realtions:

(4.5)A2=−a2,A3=2a22−a3,andA4=−5a23+5a2a3−a4.

On subsituting the values of a_i from (4.4) into (4.5) and a simple calculation yields

A2=−B14(1+α)p1,A3=−B16(1+2α)p2+(B128(1+α)2−B212(1+2α)+B112(1+2α))p12.

In (4.2), on taking c1=p12, c2=12(p2−p122), c3=18(p13−4p1p2+4p3) and so on we get,

(4.6)2(1+α)a2=B1c1,3(1+2α)a3=B1c2+B2c12,and4(1+3α)a4=B1c3+2B2c1c2+B3c13.

On substituting the values of a_i from (4.6) in (4.5), we obtain

A4=−B14(1+3α)c3+(5B126(1+α)(1+2α)−B24(1+3α))c1c2+(−B34(1+3α)+5B1B26(1+α)(1+2α)+−5B138(1+α)3)c13.

Since |p₁| ≤ 2, we have

|A2|≤B12|1+α|.

Consider

|A3|=B16|1+2α||p2−(3B1(1+2α)4(1+α)2−B22B1+12)p12|.

Then by Lemma 4.1, we get the desired estimate. Using Lemma 3.2 for A₄, we obtain

|A4|=|−B14(1+3α)c3+(−B22(1+3α)+5B126(1+α)(1+2α))c1c2+(−B34(1+3α)+5B1B26(1+α)(1+2α)−5B138(1+α)3)c13|=B14(1+3α)|c3+c1c2q1+c13q2|≤B14(1+3α)H(q1;q2),

where

q1=2(B2B1−5B1(1+3α)3(1+α)(1+2α))

and

q2=B3B1−10B2(1+3α)3(1+α)(1+2α)+5B12(1+3α)2(1+α)3.

The following corollaries are an immediate consequence of the Theorem 4.1 for ϕ(z) = e^z, ϕ(z)=1+z and ϕ(z) = (1 + z)/(1 – z), respectively.

Corollary 4.1.1

Letα∈ℂsuch that Re α ≥ 0 and ϕ(z) = e^z. If the functionf(z)=z+∑n=2∞anzn∈ℛ(α,ϕ)andf−1(ω)=ω+∑n=2∞Anωnfor all ω in some neighbourhood of the origin, then

|A2|≤12|1+α|,
|A3|≤13|1+2α|max{1,|3(1+2α)2(1+α)2−12|}
|A4|≤14(1+3α)H(q1;q2),

where α is real and H(q₁; q₂) is given inLemma 3.2such that

q1=−7+3α(−7+2α)3(1+α)(1+2α)

and

q2=16−5(1+3α)3(1+α)(1+2α)+5(1+3α)2(1+α)3.

COROLLARY 4.1.2.

Suppose that the functionf(z)=z+∑n=2∞anzn∈ℛ(α,ϕ), where Re α ≥ 0 andϕ(z)=1+zandf−1(ω)=ω+∑n=2∞Anωnfor all ω in some neighbourhood of the origin, then

|A2|≤14|1+α|,
|A3|≤16|1+2α|max{1,|3(1+2α)4(1+α)2+14|},
|A4|≤18(1+3α)H(q1;q2),

where α is real and H(q₁; q₂) is given inLemma 3.2such that

q1=−12−5(1+3α)3(1+α)(1+2α)

and

q2=18+5(1+3α)12(1+α)(1+2α)+5(1+3α)8(1+α)3.

5. Hankel determinant

The problem involving coefficient bounds have attracted the interest of many authors in particular to second Hankel determinants and Fekete Szegö functional. The coefficient functional |a3−μa22| where μ is a complex number is called the Fekete Szegö functional. The Fekete Szegö problem involves maximizing the functional |a3−μa22|. Some authors have investigated the Fekete Szegö problem for the coefficients of inverse function [2, 25, 38]. The Hankel determinant |H2(1)|=|a3−a22| is a particular case of the Fekete Szegö functional and H2(2)=|a2a4−a23| is called the second Hankel determinants. In 2013, Lee et. al. [17] obtained the bounds for the second Hankel determinant for the unified class of Ma-Minda starlike and convex functions. The authors [38] estimated the bounds on second Hankel determinant for the class of strongly convex functions of order α using the inverse coefficients. One may refer to [8, 9, 14, 15, 27] for more details. In the present section, we shall determine the Fekete Szegö functional for the inverse coefficient.

THEOREM 5.1

Supposeα∈ℂsuch that Re α ≥ 0 and the function ϕ defined by (1.2). Letf∈ℛ(α,ϕ)andf−1(ω)=ω+∑n=2∞Anωnfor all ω in some neighbourhood of the origin. Then for any complex number μ, we have

|A3−μA22|≤B13|1+2α|max{1,|3B1(1+2α)4(1+α)2(μ−2)+B2B1|}.

Proof. In view of euqations (4.4) and (4.5), we get

|A3−μA22|=|−B16(1+2α)p2+((B128(1+α)2+B1−B212(1+2α))−μB1216(1+α)2)p12|=|B16(1+2α)(p2+(−3B1(1+2α)4(1+α)2+B22B1−12+μ3B1(1+2α)8(1+α)2)p12)|=|B16(1+2α)(p2−νp12)|,

where ν=3B1(1+2α)8(1+α)2(2−μ)−B22B1+12. By Lemma 4.1, we get the required result.

THEOREM 5.2.

Letα∈ℂsuch that Re α ≥ 0 and the function ϕ defined by (1.2). Suppose function f inℛ(α,ϕ)andf−1(ω)=ω+∑n=2∞Anωnfor all ω in some neighbourhood of the origin.

If B₁, B₂and B₃satisfy the conditions
4d2≤d3,d1≤B1|1+α|9|(1+2α)2|,
then
|A2A4−A32|≤B129|(1+2α)2|.
If B₁, B₂and B₃satisfy the conditions
4d2≥d3,d1−d22−B116|1+3α|≥0
or
4d2≤d3,d1≥B1|1+α|9|(1+2α)2|,
then
|A2A4−A32|≤B1d1|1+α|.
If B₁, B₂and B₃satisfy the conditions
4d1>d3,d1−d22−B116|1+3α|≤0,
then
|A2A4−A32|≤B116|1+α|(16B1|1+α|9|(1+2α)2|d1−B1|1+3α|d2−4d22−B1216|(1+3α)2|)d1−d2−B18|1+3α|+B1|1+a|9|(1+2α)2|,
where
d1=B1316|(1+α)3|+B22|1+α|9B1|(1+2α)2|+B1|B2|12|1+α||1+2α|+|B3|8|1+3α|,d2=B1212|1+α||1+2α|+|B2|4|1+3α|+2|B2||1+α|9|(1+2α)2|,
and
d3=8B1|1+α|9|(1+2α)2|−B12|1+3α|.

In the proof of this result, the following lemma is needed.

LEMMA 5.1

([18: Lemma 2, p. 254]). Ifp(z)=1+∑n=1∞pnzn∈P, then

2p2=p12+x(4−p12),4p3=p13+2p1(4−p12)x−p1(4−p12)x2+2(4−p12)(1−|x|2)z

for some x and z such that |x| ≤ 1 and |z| ≤ 1.

Remark 3.

For real numbers P, Q and R, a standard computation gives

(5.1)max0≤t≤4(Pt2+Qt+R)={R,Q≤0,P≤−Q/4,16P+4Q+R,Q≥0,P≥−Q/8orQ≤0,P≥−Q/4,4PR−Q24P,Q>0,P≤−Q/8.

Proof of Theorem 5.2. It follows from equations (4.4) and (4.5) that

A2A4−A32=−B1p14(1+α)(−B18(1+3α)p3+(5B1224(1+α)(1+2α)+B1−B28(1+3α))p1p2+(−5B136(1+α)3−5B1(B2−B1)48(1+α)(1+2α)−B1−2B2+B332(1+3α))p13)−((B128(1+α)2+(B1−B2)12(1+2α))p12−B1p26(1+2α)+)2=p14(B14256(1+α)4+B12(B1−B2)192(1+α)2(1+2α)+B1(B1−2B2+B3)128(1+α)(1+3α)−(B1−B2)2144(1+2α)2)+B12p1p332(1+α)(1+3α)−B12p2236(1+2α)2+p12p2(−B1396(1+α)2(1+2α)−B1(B2−B1)32(1+α)(1+3α)+B1(B1−B2)36(1+2α)2).

In view of Lemma 5.1, we obtain

A2A4−A32=B116(1+α)[p14(B1316(1+α)3−B22(1+α)9B1(1+2α)2−B1B212(1+α)(1+2α)+B38(1+3α))+2p12(4−p12)x(−B1224(1+α)(1+2α)+B28(1+3α)−B2(1+α)9(1+2α)2)+(4−p12)x2p12(−B18(1+3α))−(4−p12)2x2(B1(1+α)9(1+2α)2)+2p1(4−p12)(1−|x|2)zB18(1+3α)].

Since |p_i| ≤ 2 and it can be assumed that p₁ > 0 and thus we get that p₁ ∈ [0, 2]. Letting p₁ = p and |x| = γ in the above expression, we get

|A2A4−A32|≤B116|1+α|[p4(B1316|(1+α)3|+B22|1+α|9B1|(1+2α)2|+B1|B2|12|1+α||1+2α|+|B3|8|1+3α|)+2p2(4−p2)γ(B1224|1+α||1+2α|+|B2|8|1+3α|−|B2||1+α|9|(1+2α)2|)+(4−p2)γ2p2(B18|1+3α|)+(4−p2)2γ2(B1|1+α|9|(1+2α)2|)+2p(4−p2)(1−γ2)|z|B18|1+3α|].=:G(p,γ).

Let p be fixed. Using the first derivative test in the region Ω = {(p, γ) : 0 ≤ p ≤ 2, 0 ≤ 7 ≤ 1} we get that G(p, γ) is an increasing function of γ where γ ∈ [0, 1]. Thus for fixed p ∈ [0, 2], we obtain maxo≤γ≤1G(p,γ)=G(p,1)=:F(p), where

F(p)=B116|1+α|[p4(B1316|(1+α)3|+B22|1+α|9B1|(1+2α)2|+B1|B2|12|1+α||1+2α|+|B3|8|1+3α|−B1212|1+α||1+2α|−|B2|4|1+3α|−2|B2||1+α|9|(1+2α)2|−B18|1+3α|+B1|1+α|9|(1+2α)2|)+p2(B123|1+α||1+2α|+|B2||1+3α|+8|B2||1+α|9|(1+2α)2|+B12|1+3α|−8B1|1+α|9|(1+2α)2|)+16B1[1+α]9|(1+2α)2|].

Let

P=B116|1+α|[(B1316|(1+α)3|+B139B1|(1+2α)2|+B1|B2|12|1+α||1+2α|+|B3|8|1+3α|)−(B1212|1+α||1+2α|+|B2|4|1+3α|+2|B2||1+α|9|(1+2α)2|)+(−B18|1+3α|+B1|1+α|9|(1+2α)2|)],Q=B116|1+α|[4(B1212|1+α||1+2α|+B24|1+3α|+2|B2||1+α|9|(1+2α)2|)−(8B1|1+α|9|(1+2α)2|−B12|1+3α|)],R=B129|(1+2α)2|,andt=p2.

Then F(t) = Pt² + Qt + R. Using the inequality (5.1), we get the required result.

(Communicated by Stanislawa Kanas)

Acknowledgement

The authors are grateful to the referees for their helpful suggestions and insights that helped to improve quality and clarity of this manuscript.

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Received: 2020-07-13

Accepted: 2021-01-23

Published Online: 2022-02-16

Published in Print: 2022-02-16

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Articles in the same Issue

https://doi.org/10.1515/ms-2022-0006

Keywords for this article

Differential subordination; growth theorem; distortion theorem; logarithmic coefficient; inverse coefficient; Hankel determinant

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