A Note on the Class of Functions with Bounded Turning

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.

1. Introduction

Let 𝑈∶={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 form𝑓(𝑧)=𝑎+𝑎_𝑛𝑧^𝑛+𝑎_𝑛+1𝑧^𝑛+1+⋯. (1.1) Further, let𝒜∶=𝑓∈ℋ∶𝑓(0)=𝑓^′(0)−1=0, (1.2) 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 conditionℜ𝑧𝑓^′(𝑧)𝑓(𝑧)>0,𝑓(𝑧)≠0 (1.3) in 𝑈. Also, it is known that 𝑧𝑓^′(𝑧) is starlike if and only if 𝑓 is convex, which is characterized by the analytic conditionℜ1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)>0,𝑓^′(𝑧)≠0 (1.4) 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 by𝑓(𝑧)∗𝑔(𝑧)=𝑧+^∞_𝑛=2𝑎_𝑛𝑏_𝑛𝑧^𝑛, (1.5) where 𝑎_𝑛 and 𝑏_𝑛 are the coefficients of 𝑓 and 𝑔, respectively.

The pre-Schwarzian derivative 𝑇_𝑓 of 𝑓 is defined by𝑇_𝑓(𝑧)=𝑓^′′(𝑧)𝑓^′(𝑧) (1.6) with the norm𝑇_𝑓=sup_𝑧∈𝑈𝑇_𝑓1−|𝑧|². (1.7) It is known that ‖𝑇_𝑓‖<∞ if and only if 𝑓 is uniformly locally univalent. It is also known that ‖𝑇_𝑓‖≤6 for 𝑓∈𝒮 and that ‖𝑇_𝑓‖≤4 for 𝑓∈𝒦. Moreover, it is showed that, when |𝑇_𝑓|≤3.05𝑓 is univalent in 𝑈 and when |𝑇_𝑓|≤2.28329, 𝑓 is starlike in 𝑈 (see [1]). Recently, the sharp norm estimates for well-known integral operators are determined (see [2–4]).

For 0≤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]):^𝑘𝑓^′(𝑧)≺1+𝐶𝑧1+𝐷𝑧,𝑘≥1. (1.8) 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 numbers 𝜖>0 and 𝛼>0, let us consider the class 𝐵(𝑝_𝜖):𝐵𝑝_𝜖=𝑓∈𝒜∶𝑓^′(𝑧)^𝛼−1<ℜ𝑓^′(𝑧)^𝛼−𝜖,𝑧∈𝑈. (1.9) It is easy to see that 𝑓∈𝐵(𝑝_𝜖) if and only if𝑓^′(𝑧)≺𝑝_𝜖𝜖∶=1−𝑘𝑧^1/𝛼,𝑧∈𝑈,𝑘≥2. (1.10) For the development of the paper we need to define a set 𝒬 by all points on the right half-plane as follows:𝒬=𝑤∈ℂ∶𝑤^𝛼−1<ℜ𝑤^𝛼−𝜖,𝛼>0. (1.11) Thus it is easy to verify that its boundary satisfies the equation of a parabola𝑣²=2𝑢(1−𝜖)−(1−𝜖)²,(𝑤=𝑢+𝑖𝑣). (1.12)

2. Main Results

First, our result is in the following form.

Theorem 2.1. A function 𝑓∈𝐵(𝑝_𝜖), 𝜖>0, if and only if there exists an analytic function 𝑝, 𝑝(𝑧)≺𝑝_𝜖(𝑧)𝜖∶=1−𝑘𝑧^1/𝛼,𝑘≥2, (2.1) such that 𝑓(𝑧)=^𝑧₀𝑝(𝑡)𝑑𝑡,𝑝(0)=1. (2.2) Moreover, if function 𝑓_𝜖∈𝐵(𝑝_𝜖) takes the form 𝑓_𝜖(𝑧)=(2/𝜖)−(2/𝜖)[1−(𝜖𝑧/2)]^{(2+𝛼)/𝛼}(),2+𝛼/𝛼 (2.3) then the subordination relation 𝑓(𝑧)𝑧≺𝑓_𝜖(𝑧)𝑧,𝑧∈𝑈 (2.4) holds.

Proof. Let 𝑓∈𝐵(𝑝_𝜖), and let 𝑝(𝑧)=𝑓^′(𝑧)≺𝜖1−𝑘𝑧^1/𝛼. (2.5) Integrating this equation yields (2.2). If 𝑓 is given in (2.2) with an analytic function 𝑝(𝑧)≺𝑝_𝜖(𝑧), then by a differentiation of (2.2) we obtain that 𝑓^′(𝑧)=𝑝(𝑧); therefore, 𝑓^′(𝑧)≺𝜖1−𝑘𝑧^1/𝛼, (2.6) and consequently 𝑓∈𝐵(𝑝_𝜖).
Now we proceed to prove that 𝑓_𝜖∈𝐵(𝑝_𝜖). For this purpose we will show that the set 𝒬_𝜖∶=𝑤∈ℂ∶𝑤^𝛼/2−1<ℜ(𝑤^𝛼)−𝜖−1,ℜ𝑤^𝛼>𝜖⊂𝒬. (2.7) Let 𝑤∈𝒬_𝜖. Then 𝑤^𝛼/2−1<ℜ(𝑤^𝛼)−𝜖−1⟹𝑤^𝛼/2+1<ℜ(𝑤^𝛼)−𝜖+1. (2.8) Multiplying these inequalities, we obtain 𝑤^𝛼−1<ℜ𝑤^𝛼−𝜖−1<ℜ𝑤^𝛼−𝜖. (2.9) Therefore, 𝑤∈𝒬.
Define a function 𝑞_𝜖(𝑧), 𝑧∈𝑈, as follows: 𝑞_𝜖(𝑧)∶=1−𝜖𝑧2^2/𝛼. (2.10) We suppose that 𝑤^𝛼/2∶=𝑞_𝜖(𝑧)^𝛼/2=1−𝜖𝑧2, (2.11) and thus 𝑞_𝜖(𝑈)=𝑤∈ℂ∶𝑤^𝛼/2−1<ℜ(𝑤^𝛼)−𝜖−1,ℜ𝑤^𝛼>𝜖⊂𝒬. (2.12) Hence, 𝑞_𝜖(𝑧)≺𝑝_𝜖(𝑧). Putting 𝑞_𝜖(𝑧) in (2.2) implies (2.3). To prove the subordination relation (2.4), first we show that 𝑓_𝜖(𝑧)/𝑧 is a convex function. We observe that 𝑓_𝜖(𝑧)=(2/𝜖)−(2/𝜖)[1−(𝜖𝑧/2)]^{(2+𝛼)/𝛼}()𝜖2+𝛼/𝛼=𝑧−𝑧2𝛼²+⋯=𝑧+^∞_𝑛=2𝜆(𝛼,𝜖)𝑧^𝑛∈𝒜. (2.13) Let us consider the function 𝐹_𝜖(𝑧)=−2𝛼𝜖𝑓_𝜖(𝑧)𝑧−1∈𝒜. (2.14) Computations give 𝐹^′_𝜖(𝑧)=−2𝛼𝜖𝑓^′_𝜖(𝑧)𝑧−𝑓_𝜖(𝑧)𝑧²,𝐹_𝜖^′′(𝑧)=−2𝛼𝜖𝑧𝑓_𝜖^′′(𝑧)−𝑓^′_𝜖(𝑧)𝑧²−𝑧²𝑓^′_𝜖(𝑧)−2𝑧𝑓_𝜖(𝑧)𝑧⁴. (2.15)(2.16) Also, a calculation implies that 𝑓_𝜖(𝑧)=(2/𝜖)−(2/𝜖)[1−(𝜖𝑧/2)]^{(2+𝛼)/𝛼}(2+𝛼)/𝛼,𝑓^′_𝜖(𝑧)=1−𝜖𝑧2^2/𝛼,𝑓_𝜖^′′(𝑧)=−𝜖𝛼1−𝜖𝑧2^{(2−𝛼)/𝛼}. (2.17) The aim is to show that 1+(𝑧𝐹_𝜖^′′(𝑧)/𝐹^′_𝜖(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 ℜ1+𝑧𝐹_𝜖^′′(𝑧)𝐹^′_𝜖(𝑧)𝑧=ℜ²𝑓_𝜖^′′(𝑧)𝑧𝑓^′_𝜖(𝑧)−𝑓_𝜖(𝑧)−1>0. (2.18) Consequently, we obtain that 𝐹_𝜖∈𝒦, and therefore 𝑓_𝜖(𝑧)/𝑧 is a convex function.
Now by using the fact that if 𝐹,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 class 𝐵(𝑝_𝜖), 𝜖∈(0,1] of functions 𝑓∈𝒜 of bounded turning which satisfies the relation 𝑓^′(𝑧)≺𝑝_𝜖𝜖∶=1−𝑘𝑧^1/𝛼,𝛼>1,𝑘≥2. (2.19) Then, 𝑓≤𝜖(1+𝜖/𝑘)𝑘𝛼. (2.20)

Proof. Let 𝑓∈𝐵(𝑝_𝜖), and let 𝑃_𝑓∶=𝑓^′(𝑧). Then, there exists an analytic function 𝑤∶𝑈→𝑈 with 𝑤(0)=0 and 𝑃_𝑓=𝑝_𝜖𝜖∘𝑤=1−𝑘𝑤^1/𝛼. (2.21) Define a function 𝐹∈𝒜 such that 𝑃_𝐹=−𝑝_𝜖, that is, 𝐹^′(𝑧)𝜖=−1−𝑘𝑧^1/𝛼, (2.22) and thus 𝐹(𝑧)=−^𝑧₀𝑝_𝜖(𝑡)𝑑𝑡=𝑘𝛼𝜖()𝜖𝛼+11−𝑘𝑧^{(1+𝛼)/𝛼}−𝑘𝛼𝜖().𝛼+1 (2.23) We proceed to determine the quantities 𝑇_𝐹(|𝑧|) and 𝑇_𝑓(𝑧). Logarithmic differentiation of (2.22) yields that 𝑇_𝐹𝑧=𝐹(||)^′′(|𝑧|)𝐹^′𝑧=𝜖(||)((𝑧,𝑘𝛼1−𝜖/𝑘)||) (2.24) and the logarithmic differentiation of (2.21) gives 𝑇_𝑓(𝑧)=𝑓^′′(𝑧)𝑓^′(𝑧)𝜖=−𝑤𝑘𝛼^′(𝑧)1−𝜖/𝑘𝑤(𝑧),𝑧≠0. (2.25) Thus, by triangle inequality and the Schwarz-Pick lemma, we obtain 𝑇_𝑓(𝑧=𝜖)𝑤𝑘𝛼^′(𝑧)()𝑤(𝑧)≤𝜖1−𝜖/𝑘|𝑤(𝑧𝑘𝛼1−)|²1−|𝑧|²()𝑤(𝑧≤𝜖(|1−𝜖/𝑘)|)|𝑤(𝑧𝑘𝛼1−)|(|𝑧()𝑤(𝑧≤𝜖1−|)(|1−𝜖/𝑘)|)1𝑘𝛼(|𝑧≤𝜖1−|)1𝑘𝛼(1−(𝜖/𝑘)|𝑧|)=𝑇_𝐹(|𝑧|). (2.26) Consequently, we have 1−|𝑧|²𝑇_𝑓(𝑧)=1−|𝑧|²𝑇_𝐹(|𝑧|). (2.27) Therefore, ‖𝑓‖≤‖𝐹‖, and this inequality is sharp. Thus, to determine the upper estimate of 𝑓∈𝐵(𝑞_𝜖) it is enough to compute ‖𝐹‖. Letting 𝑡=|𝑧|, we have sup1−𝑡²𝑇_𝐹(|𝑧|)=sup1−𝑡²𝜖(()𝑡)𝜖𝑘𝛼1−𝜖/𝑘≤sup1−²𝑘²𝑡²𝜖(()𝑡)𝜖(()𝑡)𝑘𝛼1−𝜖/𝑘,𝑘>𝜖=sup1+𝜖/𝑘.𝑘𝛼 (2.28) Hence, we obtain (2.20).

By applying Jack’s lemma, we pose the sufficient conditions for convex functions 𝑓 belonging to the subclasses 𝐵(𝑝_𝜖) and 𝐵(𝑞_𝜖)=(1+𝜖𝜔(𝑧))^1/𝛼.

Lemma 2.3 ( see [11]). Let 𝑤 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 𝑧₀𝑤^′𝑧₀=𝑚𝑤𝑧₀, (2.29) where 𝑚 is a real number and 𝑚≥1.

Theorem 2.4. Assume that 𝜖∈[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(𝑝_𝜖).

Proof. Let 𝑓∈𝒦(𝛼−𝜖(1+𝛼)/𝛼(1−𝜖)), that is, ℜ1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)>𝑘𝛼−𝜖(1+𝛼)𝑘𝛼(1−𝜖/𝑘),𝑧∈𝑈. (2.30) From the proof of Theorem 2.2, we have ℜ𝑧𝑓^′′(𝑧)𝑓^′(𝑧)𝜖+1=ℜ1−𝑘𝛼𝑧𝑤^′(𝑧)()𝑤(𝑧)>()1−𝜖/𝑘𝑘𝛼−𝜖1+𝛼𝑘𝛼(1−𝜖/𝑘),𝑧∈𝑈, (2.31) where 𝑤 is analytic in 𝑈 and satisfies 𝑤(0)=0 and 𝑓^′(𝑧)=𝜖1−𝑘𝑤(𝑧)^1/𝛼. (2.32) Suppose that there exists a point 𝑧₀∈𝑈 such that max_{|𝑧|≤𝑧0}|𝑤(𝑧)|=𝑤𝑧₀=1. (2.33) Then, on using Lemma 2.3 and letting 𝑤(𝑧₀)=𝑒^𝑖𝜃 and 𝑧₀𝑤^′(𝑧₀)=𝑚𝑒^𝑖𝜃, 𝑚≥1 yields ℜ𝑧₀𝑓^′′𝑧₀𝑓^′𝑧₀𝜖+1=ℜ1−𝑧𝑘𝛼⁰𝑤^′𝑧₀1−𝜖/𝑘𝑤𝑧₀𝜖=ℜ1−𝑘𝛼𝑚𝑒^𝑖𝜃1−𝜖/𝑘𝑒^𝑖𝜃𝜖≤ℜ1−𝑒𝑘𝛼1−𝜖/^𝑖𝜃𝑘𝑒^𝑖𝜃, (2.34) which contradicts hypothesis (2.30). Therefore, we conclude that |𝑤(𝑧)|<1 for all 𝑧∈𝑈, that is, 𝑓∈𝐵(𝑝_𝜖).

Theorem 2.5. Assume that 𝜖∈(0,1] and 𝛼>1. If 𝑓∈𝒜 satisfies ℜ𝑧𝑓^′′(𝑧)𝑓^′(𝑧)<𝜖()+1𝜖/𝑘+2𝑘𝛼(𝜖/𝑘+1),𝑧∈𝑈, (2.35) then 𝑓∈𝐵(𝑞_𝜖).

Proof. Define a function 𝜔(𝑧) by 𝑓^′(𝑧)=𝜖1+𝑘𝜔(𝑧)^1/𝛼. (2.36) Then, 𝜔 is analytic in 𝑈 and satisfies 𝜔(0)=0. It follows that ℜ𝑧𝑓^′′(𝑧)𝑓^′(𝑧)=𝜖+1ℜ𝑘𝛼𝑧𝜔^′(𝑧)+(𝜖/𝑘)𝜔(𝑧)+1()𝜔(𝑧)<𝜖()1+𝜖/𝑘𝜖/𝑘+2().𝑘𝛼𝜖/𝑘+1 (2.37) In the same manner of Theorem 2.4, we find that ℜ𝑧₀𝑓^′′𝑧₀𝑓^′𝑧₀=𝜖+1𝛼ℜ𝑧₀𝜔^′𝑧₀+𝜖𝜔𝑧₀+11+𝜖𝜔𝑧₀=𝜖ℜ𝑘𝛼𝑚𝑒^𝑖𝜃+(𝜖/𝑘)𝑒^𝑖𝜃+11+(𝜖/𝑘)𝑒^𝑖𝜃≥𝜖(𝜖/𝑘+2)(),𝑘𝛼𝜖/𝑘+1 (2.38) which contradicts hypothesis (2.35). Therefore, we conclude that |𝜔(𝑧)|<1 for all 𝑧∈𝑈, that is, 𝑓∈𝐵(𝑞_𝜖).

Corollary 2.6. Let the assumptions of Theorem 2.5 hold. Then, 𝑓 is strongly close to convex of order 1/𝛼.

Proof. Since 𝑓∈𝐵(𝑞_𝜖), there exists an analytic function 𝜓∈𝑈 such that 𝜓(0)=0, |𝜓(𝑧)|<1 and 𝑓^′(𝑧)^𝛼𝜖=1+𝑘𝜓(𝑧). (2.39) But 𝜓(𝑧=𝑓)^′(𝑧)^𝛼−1𝜖/𝑘<1, (2.40) which implies 𝑓^′(𝑧)^𝛼<𝜖−1𝑘<1 (2.41) 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(𝑝_𝜖); Theorem 2.4). Moreover, some classes of bounded turning functions can embed in the class of close-to-convex functions (𝐵(𝑞_𝜖)⊂𝒦ℒ; Corollary 2.6). Hence, we have the following inclusion relation:𝒦⊂𝐵⊂𝒦ℒ. (3.1)

Acknowledgments

The authors would also like to thank the referee for valuable remarks and suggestions on the previous version of the paper. This research has been funded by University of Malaya, under Grant no. RG208-11AFR.