# A Note on the Class of Functions with Bounded Turning

Authors: Rabha W. Ibrahim; Adem Kılıçman
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We consider subclasses of functions with bounded turning for normalized analytic functions in the unit disk. The geometric representation is introduced, some subordination relations are suggested, and the upper bound of the pre-Schwarzian norm for these functions is computed. Moreover, by employing Jack's lemma, we obtain a convex class in the class of functions of
bounded turning and relations with other classes are posed.


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## Body

## 1. Introduction

LetU:={z:|z|<1} be the unit disk in the complex plane ℂ, and let ℋ denote the space of all analytic functions on U. Here, we suppose that ℋ is a topological vector space endowed with the topology of uniform convergence over compact subsets of U. Also, for a∈ℂ and n∈ℕ, let ℋ[a,n] be the subspace of ℋ consisting of functions of the form(1.1)f(z)=a+anzn+an+1zn+1+⋯.
Further, let(1.2)A:={f∈H:f(0)=f′(0)-1=0},
and let 𝒮 denote the class of univalent functions in 𝒜. A function f∈𝒜 is called starlike if f(U) is a starlike domain with respect to the origin, and the class of univalent starlike functions is denoted by S*. It is called convex 𝒞 if f(U) is a convex domain. Each univalent starlike function f is characterized by the analytic condition(1.3)R(zf′(z)f(z))>0,f(z)≠0
in U. Also, it is known that zf′(z) is starlike if and only if f is convex, which is characterized by the analytic condition(1.4)R(1+zf′′(z)f′(z))>0,f′(z)≠0
in U. Let f∈ℋ, and let g be a univalent function in U, with f(0)=g(0). Then, we say that f is subordinate to g (or g is superordinate to f), denoted by f(z)≺g(z), if f(U)⊂g(U). For two functions f,g∈𝒜, the Hadamard product is defined by(1.5)f(z)*g(z)=z+∑n=2∞anbnzn,
where an and bn are the coefficients of f and g, respectively.The pre-Schwarzian derivativeTf of f is defined by(1.6)Tf(z)=f′′(z)f′(z)
with the norm(1.7)‖Tf‖=sup⁡z∈U|Tf|(1-|z|2).
It is known that ∥Tf∥<∞ if and only if f is uniformly locally univalent. It is also known that ∥Tf∥≤6 for f∈𝒮 and that ∥Tf∥≤4 for f∈𝒦. Moreover, it is showed that, when |Tf|≤3.05f is univalent in U and when |Tf|≤2.28329, f is starlike in U (see [1]). Recently, the sharp norm estimates for well-known integral operators are determined (see [2–4]).For0≤ν<1, let B(ν) denote the class of functions f of the form (2.2) so that ℜ{f′}>ν in U. The functions in B(ν) are called functions of bounded turning. By the Nashiro-Warschowski theorem, the functions in B(ν) are univalent and also close to convex in U. It is well known that B(ν)⊈S* and S*⊈B(ν). In [5], Mocanu obtained a subclass of S*, which is contained in B(ν). Recently, Tuneski generalized the class of convex functions with bounded turning (see [6]):(1.8)f′(z)k≺1+Cz1+Dz,k≥1.
Different studies of the class of bounded turning functions can be found in [7–10].In this note we pose the following subclass of bounded turning functions in the the unit disk: for given numbersϵ>0 and α>0, let us consider the class B(pϵ):(1.9)B(pϵ)={f∈A:|(f′(z))α-1|<R(f′(z))α-ϵ,z∈U}.
It is easy to see that f∈B(pϵ) if and only if(1.10)f′(z)≺pϵ:=(1-ϵkz)1/α,z∈U,k≥2.
For the development of the paper we need to define a set 𝒬 by all points on the right half-plane as follows:(1.11)Q={w∈C:|wα-1|<R(wα)-ϵ,α>0}.
Thus it is easy to verify that its boundary satisfies the equation of a parabola(1.12)v2=2u(1-ϵ)-(1-ϵ)2,(w=u+iv).


## 2. Main Results

First, our result is in the following form.Theorem 2.1.
A functionf∈B(pϵ), ϵ>0, if and only if there exists an analytic function p,
(2.1)p(z)≺pϵ(z):=(1-ϵkz)1/α,k≥2,
such that
(2.2)f(z)=∫0zp(t)dt,p(0)=1.
Moreover, if function fϵ∈B(pϵ) takes the form
(2.3)fϵ(z)=(2/ϵ)-(2/ϵ)[1-(ϵz/2)](2+α)/α(2+α)/α,
then the subordination relation
(2.4)f(z)z≺fϵ(z)z,z∈U
holds.Proof.
Letf∈B(pϵ), and let
(2.5)p(z)=f′(z)≺(1-ϵkz)1/α.
Integrating this equation yields (2.2). If f is given in (2.2) with an analytic function p(z)≺pϵ(z), then by a differentiation of (2.2) we obtain that f′(z)=p(z); therefore,
(2.6)f′(z)≺(1-ϵkz)1/α,
and consequently f∈B(pϵ).
Now we proceed to prove thatfϵ∈B(pϵ). For this purpose we will show that the set
(2.7)Qϵ:={w∈C:|wα/2-1|<R(wα)-ϵ-1,R(wα)>ϵ}⊂Q.
Let w∈𝒬ϵ. Then
(2.8)|wα/2-1|<R(wα)-ϵ-1⟹|wα/2+1|<R(wα)-ϵ+1.
Multiplying these inequalities, we obtain
(2.9)|wα-1|<R(wα)-ϵ-1<R(wα)-ϵ.
Therefore, w∈𝒬.
Define a functionqϵ(z), z∈U, as follows:
(2.10)qϵ(z):=[1-ϵz2]2/α.
We suppose that
(2.11)wα/2:=[qϵ(z)]α/2=1-ϵz2,
and thus
(2.12)qϵ(U)={w∈C:|wα/2-1|<R(wα)-ϵ-1,R(wα)>ϵ}⊂Q.
Hence, qϵ(z)≺pϵ(z). Putting qϵ(z) in (2.2) implies (2.3). To prove the subordination relation (2.4), first we show that fϵ(z)/z is a convex function. We observe that
(2.13)fϵ(z)=(2/ϵ)-(2/ϵ)[1-(ϵz/2)](2+α)/α(2+α)/α=z-ϵ2αz2+⋯=z+∑n=2∞λ(α,ϵ)zn∈A.
Let us consider the function
(2.14)Fϵ(z)=-2αϵ[fϵ(z)z-1]∈A.
Computations give
(2.15)Fϵ′(z)=-2αϵ[fϵ′(z)z-fϵ(z)z2],(2.16)Fϵ′′(z)=-2αϵ[zfϵ′′(z)-fϵ′(z)z2-z2fϵ′(z)-2zfϵ(z)z4].
Also, a calculation implies that
(2.17)fϵ(z)=(2/ϵ)-(2/ϵ)[1-(ϵz/2)](2+α)/α(2+α)/α,fϵ′(z)=[1-ϵz2]2/α,fϵ′′(z)=-ϵα[1-ϵz2](2-α)/α.
The aim is to show that 1+(zFϵ′′(z)/Fϵ′(z)) has positive real part in the unit disk. For ℜ(z)>0 and a suitable choice of 0<ϵ≤1 such that 0<ℜ(1-ϵz/2)<1 and by using (2.17), we have
(2.18)R(1+zFϵ′′(z)Fϵ′(z))=R(z2fϵ′′(z)zfϵ′(z)-fϵ(z)-1)>0.
Consequently, we obtain that Fϵ∈𝒦, and therefore fϵ(z)/z is a convex function.
Now by using the fact that ifF,G∈𝒦, satisfy f≺F and g≺G, then f*g≺F*G and k(z)=z/(1-z) is a convex function, immediately we establish (2.4). This completes the proof.Next we consider another class of functions of bounded turning. We will estimate the upper bound of these functions by using the pre-Schwarzian norm.Theorem 2.2.
Consider the classB(pϵ), ϵ∈(0,1] of functions f∈𝒜 of bounded turning which satisfies the relation
(2.19)f′(z)≺pϵ:=(1-ϵkz)1/α,α>1,k≥2.
Then,
(2.20)‖f‖≤ϵ(1+ϵ/k)kα.Proof.
Letf∈B(pϵ), and let Pf:=f′(z). Then, there exists an analytic function w:U→U with w(0)=0 and
(2.21)Pf=pϵ∘w=(1-ϵkw)1/α.
Define a function F∈𝒜 such that PF=-pϵ, that is,
(2.22)F′(z)=-(1-ϵkz)1/α,
and thus
(2.23)F(z)=-∫0zpϵ(t)dt=kαϵ(α+1)(1-ϵkz)(1+α)/α-kαϵ(α+1).
We proceed to determine the quantities TF(|z|) and Tf(z). Logarithmic differentiation of (2.22) yields that
(2.24)TF(|z|)=F′′(|z|)F′(|z|)=ϵkα(1-(ϵ/k)|z|),
and the logarithmic differentiation of (2.21) gives
(2.25)Tf(z)=f′′(z)f′(z)=-ϵkα[w′(z)1-ϵ/kw(z)],z≠0.
Thus, by triangle inequality and the Schwarz-Pick lemma, we obtain
(2.26)|Tf(z)|=ϵkα|w′(z)1-(ϵ/k)w(z)|≤ϵkα1-|w(z)|2(1-|z|2)(|1-(ϵ/k)w(z)|)≤ϵkα1-|w(z)|(1-|z|)(|1-(ϵ/k)w(z)|)≤ϵkα1(1-|z|)≤ϵkα1(1-(ϵ/k)|z|)=TF(|z|).
Consequently, we have
(2.27)(1-|z|2)|Tf(z)|=(1-|z|2)TF(|z|).
Therefore, ∥f∥≤∥F∥, and this inequality is sharp. Thus, to determine the upper estimate of f∈B(qϵ) it is enough to compute ∥F∥. Letting t=|z|, we have
(2.28)sup⁡(1-t2)TF(|z|)=sup⁡(1-t2)ϵkα(1-(ϵ/k)t)≤sup⁡(1-ϵ2k2t2)ϵkα(1-(ϵ/k)t),k>ϵ=sup⁡ϵ(1+(ϵ/k)t)kα.
Hence, we obtain (2.20).By applying Jack’s lemma, we pose the sufficient conditions for convex functionsf belonging to the subclasses B(pϵ) and B(qϵ)=(1+ϵω(z))1/α.Lemma 2.3 ( see [11]).
Letw be analytic in U with w(0)=0. If |w(z)| attains its maximum value on the circle |z|=r<1 at a point z0, then
(2.29)z0w′(z0)=mw(z0),
where m is a real number and m≥1.Theorem 2.4.
Assume thatϵ∈[1/2,1) and α>ϵ/(1-ϵ/k). If f∈𝒜 is a convex function in U of order (0≤(kα-ϵ(1+α))/kα(1-ϵ/k))<1, then f∈B(pϵ).Proof.
Letf∈𝒦(α-ϵ(1+α)/α(1-ϵ)), that is,
(2.30)R{1+zf′′(z)f′(z)}>kα-ϵ(1+α)kα(1-ϵ/k),z∈U.
From the proof of Theorem 2.2, we have
(2.31)R{zf′′(z)f′(z)+1}=R{1-ϵkα[zw′(z)1-(ϵ/k)w(z)]}>kα-ϵ(1+α)kα(1-ϵ/k),z∈U,
where w is analytic in U and satisfies w(0)=0 and
(2.32)f′(z)=(1-ϵkw(z))1/α.
Suppose that there exists a point z0∈U such that
(2.33)max⁡|z|≤|z0||w(z)|=|w(z0)|=1.
Then, on using Lemma 2.3 and letting w(z0)=eiθ and z0w′(z0)=meiθ, m≥1 yields
(2.34)R{z0f′′(z0)f′(z0)+1}=R{1-ϵkαz0w′(z0)(1-ϵ/kw(z0))}=R{1-ϵkαmeiθ(1-ϵ/keiθ)}≤R{1-ϵkαeiθ(1-ϵ/keiθ)},
which contradicts hypothesis (2.30). Therefore, we conclude that |w(z)|<1 for all z∈U, that is, f∈B(pϵ).Theorem 2.5.
Assume thatϵ∈(0,1] and α>1. If f∈𝒜 satisfies
(2.35)R{zf′′(z)f′(z)+1}<ϵ(ϵ/k+2)kα(ϵ/k+1),z∈U,
then f∈B(qϵ).Proof.
Define a functionω(z) by
(2.36)f′(z)=(1+ϵkω(z))1/α.
Then, ω is analytic in U and satisfies ω(0)=0. It follows that
(2.37)R{zf′′(z)f′(z)+1}=ϵkαR{zω′(z)+(ϵ/k)ω(z)+11+(ϵ/k)ω(z)}<ϵ(ϵ/k+2)kα(ϵ/k+1).
In the same manner of Theorem 2.4, we find that
(2.38)R{z0f′′(z0)f′(z0)+1}=ϵαR{z0ω′(z0)+ϵω(z0)+11+ϵω(z0)}=ϵkαR{meiθ+(ϵ/k)eiθ+11+(ϵ/k)eiθ}≥ϵ(ϵ/k+2)kα(ϵ/k+1),
which contradicts hypothesis (2.35). Therefore, we conclude that |ω(z)|<1 for all z∈U, that is, f∈B(qϵ).Corollary 2.6.
Let the assumptions of Theorem2.5 hold. Then, f is strongly close to convex of order 1/α.Proof.
Sincef∈B(qϵ), there exists an analytic function ψ∈U such that ψ(0)=0, |ψ(z)|<1 and
(2.39)(f′(z))α=1+ϵkψ(z).
But
(2.40)|ψ(z)|=|(f′(z))α-1ϵ/k|<1,
which implies
(2.41)|(f′(z))α-1|<ϵk<1
and thus f is strongly close to convex of order 1/α.


## 3. Conclusion

It is well known that the class of bounded turning functions is not included in the class of the starlike functions and also the starlike functions cannot embed in the class of bounded turning functions. From the above, we conclude that some classes of bounded turning functions can be included in the class of convex functions (𝒦⊂B(pϵ); Theorem 2.4). Moreover, some classes of bounded turning functions can embed in the class of close-to-convex functions (B(qϵ)⊂𝒦ℒ; Corollary 2.6). Hence, we have the following inclusion relation:(3.1)K⊂B⊂KL.



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*Source: 820696-2012-04-09.xml*