# Arc Length Inequality for a Certain Class of Analytic Functions Related to Conic Regions

Authors: Wasim Ul-Haq; Muhammad Arif; Asfandyar Khan
Journal: Journal of Complex Analysis
 (2013)

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## Abstract

In our present investigation, we introduce a subclass of analytic function associated with conic regions which is a form of generalized close-to-convexity. The arc-length inequality
for a class of analytic function is well known. We derive this inequality for the newly defined class and also study some of its interesting consequences.


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## Body

## 1. Introduction

Let𝒜 denote the class of functions f:
(1)f(z)=z+∑n=2∞‍anzn,
which are analytic in the unit disc 𝒰={z:|z|<1}. Let 𝒮 denote the class of all functions in 𝒜 which are univalent. Also let 𝒮*, 𝒞, and 𝒦 be the well-known subclasses of 𝒜 consisting of all functions which are, respectively, of starlike, convex, and close-to-convex.Kanas and Wisniowska [1, 2] studied the classes of k-uniformly convex denoted by k-𝒰𝒞𝒱 and the corresponding class k-𝒮𝒯 related by the Alexandar type relation. Later Acu [3] considered the class k-uniformly close-to-convex denoted by k-𝒰𝒦 to be defined as
(2)k-𝒰𝒦={f(z)∈𝒜:Re(zf′(z)g(z))>k|zf′(z)g(z)-1|,22g(z)∈k-𝒮𝒯,z∈E|zf′(z)g(z)-1|};
for more detail see [4–6].In [7], the conic domain Ωk,γ with complex order is defined as
(3)Ωk,γ=γΩk+(1-γ),0<Reγ≤k+1,
where
(4)Ωk={u+iv:u>k(u-1)2+v2}.
The domain Ωk,γ is elliptic for k>1, hyperbolic when 0<k<1, parabolic for k=1, and right half plane when k=0. The functions which play the role of extremal functions for the conic regions of complex order are given as
(5)pk,γ(z)={1+(2γ-1)z1-z,k=0,1+2γπ2(log1+z1-z)2,k=1,1+2γ1-k2sinh2[(2πarccosk)arctanhz],0<k<1,1+γk2-1sin(π2R(t)∫0u(z)/t11-x21-(tx)2dx)+γk2-1,k>1,
where u(z)=(z-t)/(1-tz),  t∈(0,1), z∈E, and z is chosen such that k=cosh(πR′(t)/4R(t)),  where   R(t) is the Legendre's complete elliptic integral of the first kind and R′(t) is complementary integral of R(t), see [1, 2].LetP={p(z):p(0)=1andRep(z)>0,z∈E} be the class of functions with positive real part, and let k-P(γ) be the subclass of P containing the functions p(z), such that p(z)≺pk,γ(z). Motivated from Noor’s work [8], we extend class k-P(γ) to class k-Pm(γ),  m≥2 which is  defined as(6)k-Pm(γ)={p(z):p(0)=1and∃p1(z),p2(z)ink-P(γ)suchthatp(z)=(m4+12)p1(z)-(m4-12)p2(z),z∈E}.Note thatk-P2(γ)=k-P(γ) and 0-Pm(0)=Pm, the class introduced and studied by Pinchuk [9].We define the following class:(7)k-𝒰𝒯m(γ)={f(z)∈𝒜:f′(z)g′(z)∈P,g(z)∈k-𝒰𝒱m(γ),z∈E},
where
(8)k-𝒰𝒱m(γ)={f(z)∈𝒜:1+zf′′(z)f′(z)∈k-Pm(γ),z∈E}.
Geometrically, a function f(z)∈𝒰𝒱m(γ) means that the functional 1+(zf′′(z)/f′(z)) takes all the values in conic domain Ωk,γ and its boundary rotation is at most mπ. We note that class k-𝒰𝒯m(γ) coincides with already known classes of analytic functions by choosing special values for the involved parameters. For example, for k=0,γ=1, we have the class 𝒯m introduced and studied by Noor [10], and further along with this by taking m=2, we obtain the well-known class 𝒦 of close-to-convex functions. The purpose of this paper is to investigate some interesting properties of class k-𝒰𝒯m(γ). For this, we require the following results.Lemma 1.
A functionf∈k-𝒰𝒱m(γ) if and only if(i)
f′(z)=[f1′(z)]γ/(1+k),
f1(z)∈𝒱m,(ii)
there exist two normalized starlike functionss1(z) and s2(z) such that
(9)f′(z)=[(s1(z)/z)((k/4)+(1/2))(s2(z)/z)((k/4)-(1/2))]γ/(1+k).The above lemma can be proved by using the similar procedure as in [11]; also see [8].Lemma 2 (see [12]).
Leth∈P with z=reiθ. Then,
(10)12π∫02π|h(z)|2dθ≤1+3r21-r2.


## 2. Some Properties of the Classk-𝒰𝒯m(γ)

In this section, we provide some of the interesting properties of classk-𝒰𝒯m(γ) such as radius of convexity problem, arc length, and growth rate of its coefficients. The following theorem is readily seen when we proceed on similar lines as in [13].Theorem 3.
The functionf(z)∈k-𝒰𝒯m(γ) if and only if
(11)f′(z)=(f1(z))((m/4)+(1/2))γ(f2(z))((m/4)-(1/2))γ,
where f1(z) and f2(z) are close-to-convex functions.Theorem 4.
Letf∈k-𝒰𝒯m(ρ,γ) in E. Then, f∈C for |z|<r0, where
(12)r0=2(1+k)mγ+2k+2+[mγ+2k+2]2-4(1+k)(2γ-k-1).
This result is sharp.Proof.
We can write(13)f′(z)=g′(z)h(z),g(z)∈k-𝒰𝒱m(γ),h(z)∈P.
Using Lemma1(ii), we get
(14)f′(z)=[(s1(z)/z)((m/4)+(1/2))(s2(z)/z)((m/4)-(1/2))]γ/(1+k)h(z),
where s1 and s2 are starlike functions. Logarithmic differentiation of (14) gives us
(15)zf′′(z)f′(z)=γ1+k[-1+(m4+12)zs1′(z)s1(z)-(m4-12)zs2′(z)s2(z)]+zh′(z)h(z).
This implies that(16)1+zf′′(z)f′(z)=1+k-γ1+k+γ1+k[(m4+12)zs1′(z)s1(z)-(m4-12)zs2′(z)s2(z)]+zh′(z)h(z).
Now using distortion results for the class P, we have
(17)Re(1+zf′′(z)f′(z))≥1+k-γ1+k+γ1+k[(m4+12)1-r1+r-(m4-12)1+r1-r]-2r1-r2=(1+k-γ)(1-r2)+γ[1+r2-mr]-2r(1+k)(1+k)(1-r2).
The right hand side of (17) is positive for |z|<r0, where r0 is given by (12). The sharpness can be viewed from the function f0∈k-𝒰𝒯m(γ), given by
(18)f0′(z)=(1+z)(γ/(k+1))(((m/2)-1)+1)(1-z)(γ/(k+1))(((m/2)+1)+1),z∈E.
We note the following interesting special cases:(i)
Forγ=1, we have the radius of convexity for class k-𝒰𝒯m.(ii)
Forγ=1  and  k=0, we have the radius of convexity for class 𝒯m, proved by Noor [10].(iii)
Forγ=1, k=0  and  m=2, we have radius of convexity for close-to-convex functions which is well known.Theorem 5.
Letf∈k-𝒰𝒯m(γ) with k≥0, m≥2, and ((m+2)/(1+k))Reγ>1. Then,
(19)Lr(f)≤A(k,γ,m)(11-r)(1/2)((m+2)/(1+k))Reγ.
The exponent (1/2)((m+2)/(1+k))Reγ is sharp.Proof.
Letf∈k-𝒰𝒯m(ρ,γ). Then, there exists g(z)∈k-𝒰𝒱m(γ) such that
(20)f′(z)=g′(z)h(z),h∈P.
From the definition of k-𝒰𝒱m(γ), one can deduce that g(z)∈k-𝒰𝒱m(1) implies that g(z)∈𝒱m(k/(k+1)).
Now using (20), Lemma 1(ii), and distortion theorems for starlike functions, we have
(21)Lr(f)=∫02π|zg′(z)h(z)|dθ,g(z)∈k-𝒰𝒱m(γ),h(z)∈PLr(f)=∫02π|z(s1(z)/z)((m/4)+(1/2))(1/(1+k))γ(s2(z)/z)((m/4)-(1/2))(1/(1+k))γ||h(z)|dθLr(f)=∫02π|z1-γ(1/(k+1))(s1(z))((m/4)+(1/2))(1/(1+k))γ(s2(z))((m/4)-(1/2))(1/(1+k))γ|Lr(f)=×|h(z)|dθLr(f)≤2((m/2)-1)(1/(k+1))Reγr((m/4)+(1/2))(1/(k+1))Reγ-1Lr(f)=×∫02π|s1(z)|(1/4)((m+2)/(1+k))Reγ|h(z)|dθ.
By using Hölder's inequality, this gives
(22)Lr(f)≤2((m/2)-1)(1/(k+1))Reγr((m/4)+(1/2))(1/(k+1))Reγ-1×(12π∫02π|s1(z)|(1/2)((m+2)/(1+k))Reγdθ)1/2×(12π∫02π|h(z)|2dθ)1/2.
Since ((m+2)/(1+k))Reγ>1, therefore subordination for starlike functions and Lemma 2 give us
(23)Lr(f)≤A(k,γ,m)(11-r)(1/2)((m+2)/(1+k))Reγ.
The function F0(z)∈k-𝒰𝒯m(γ) is defined by
(24)F0′(z)=G0′(z)h0(z),
where
(25)G0′(z)=(1+z)((m/2)-1)(1/(1+k))γ(1-z)((m/2)+1)(1/(1+k))γ,h0(z)=1+z1-z
shows that the exponent is sharp.Some special choices in the above theorem give us the following interesting results.Corollary 6.
Letf∈k-𝒰𝒯2(1). Then
(26)Lr(f)≤A(k)(11-r)2/(1+k).Corollary 7.
Letf(z)∈𝒯m. Then
(27)Lr(f)≤A(m)(11-r)(m/2)+1.Coefficient Growth Problems.The problem of growth rate and asymptotic behavior of coefficients is well known. In the next results, we study these problems for class k-𝒰𝒯m(γ) by varying different parameters.Theorem 8.
Letf∈k-𝒰𝒯m(γ) with k≥0, m≥2 and ((m+2)/(1+k))Reγ>1. Then
(28)|an|=O(1)n((m/2)+1)(Reγ/(1+k))-1,(n⟶∞).
The exponent is sharp.Proof.
Withz=reiθ, Cauchy’s theorem gives us
(29)nan=12πrn∫02π|zf′(z)|dθ=12πrnLr(f),z=reiθ.
Using Theorem 5 and putting r=1-(1/n), we obtain the required result. The sharpness follows from the function F0 defined by the relation  (24).Corollary 9.
Letf∈k-𝒰𝒯m(1), and let it be of the form (1). Then, for n>3,  k≥2, one has
(30)|an|=O(1)n((m/2)+1)(1/(1+k))-1.
For k=0, in the above corollary, we have growth rate of coefficients problem for functions in class 𝒯m, and, for k=0, m=2 gives us the growth rate of coefficients for close-to-convex functions, which are well known.



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