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

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.

1. Introduction

Let 𝒜 denote the class of functions 𝑓: 𝑓(𝑧)=𝑧+^∞_𝑛=2𝑎_𝑛𝑧^𝑛, (1) 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 𝑘−𝒰𝒦=𝑓(𝑧)∈𝒜∶Re𝑧𝑓^′(𝑧)𝑔(𝑧)>𝑘𝑧𝑓^′(𝑧)𝑔(𝑧),𝑔(𝑧);−1∈𝑘−𝒮𝒯,𝑧∈𝐸 (2) for more detail see [4–6].

In [7], the conic domain Ω_𝑘,𝛾 with complex order is defined as Ω_𝑘,𝛾=𝛾Ω_𝑘+1−𝛾,0<Re𝛾≤𝑘+1, (3) where Ω_𝑘=𝑢+𝑖𝑣∶𝑢>𝑘(𝑢−1)²+𝑣². (4) The domain Ω_𝑘,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 𝑝_𝑘,𝛾(𝑧)=1+2𝛾−1𝑧1−𝑧,𝑘=0,1+2𝛾𝜋²√log1+𝑧1−√𝑧²,𝑘=1,1+2𝛾1−𝑘²sinh²2𝜋arccos𝑘arctanℎ√𝑧,𝛾0<𝑘<1,1+𝑘²𝜋−1sin(𝑡)∫2𝑅^{√𝑢(𝑧)/𝑡0}1√1−𝑥²1−(𝑡𝑥)²+𝛾𝑑𝑥𝑘²−1,𝑘>1, (5) 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].

Let 𝑃={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)≺𝑝_𝑘,𝛾(𝑧). Motivated from Noor’s work [8], we extend class 𝑘−𝑃(𝛾) to class 𝑘−𝑃_𝑚(𝛾),  𝑚≥2 which is defined as𝑘−𝑃_𝑚𝛾=𝑝(𝑧)∶𝑝(0)=1and∃𝑝₁(𝑧),𝑝₂(𝑧)𝛾𝑝(𝑧)=𝑚in𝑘−𝑃suchthat4+12𝑝₁(𝑧)−𝑚4−12𝑝₂(𝑧),𝑧∈𝐸. (6)

Note that 𝑘−𝑃₂(𝛾)=𝑘−𝑃(𝛾) and 0−𝑃_𝑚(0)=𝑃_𝑚, the class introduced and studied by Pinchuk [9].

We define the following class: 𝑘−𝒰𝒯_𝑚𝛾=𝑓(𝑧)𝑓∈𝒜∶^′(𝑧)𝑔^′(𝑧)∈𝑃,𝑔(𝑧)∈𝑘−𝒰𝒱_𝑚𝛾,𝑧∈𝐸, (7) where 𝑘−𝒰𝒱_𝑚𝛾=𝑓(𝑧)∈𝒜∶1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)∈𝑘−𝑃_𝑚𝛾,𝑧∈𝐸. (8) Geometrically, a function 𝑓(𝑧)∈𝒰𝒱_𝑚(𝛾) means that the functional 1+(𝑧𝑓^′′(𝑧)/𝑓^′(𝑧)) takes all the values in conic domain Ω_𝑘,𝛾 and its boundary rotation is at most 𝑚𝜋. We note that class 𝑘−𝒰𝒯_𝑚(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 𝒯_𝑚 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−𝒰𝒯_𝑚(𝛾). For this, we require the following results.

Lemma 1. A function 𝑓∈𝑘−𝒰𝒱_𝑚(𝛾) if and only if(i)𝑓^′(𝑧)=[𝑓^′₁(𝑧)]^{𝛾/(1+𝑘)}, 𝑓₁(𝑧)∈𝒱_𝑚, (ii)there exist two normalized starlike functions 𝑠₁(𝑧) and 𝑠₂(𝑧) such that 𝑓^′(𝑧)=𝑠₁(𝑧)/𝑧^{((𝑘/4)+(1/2))}𝑠₂(𝑧)/𝑧^{((𝑘/4)−(1/2))𝛾/(1+𝑘)}. (9)

The above lemma can be proved by using the similar procedure as in [11]; also see [8].

Lemma 2 (see [12]). Let ℎ∈𝑃 with 𝑧=𝑟𝑒^𝑖𝜃. Then, 12𝜋₀^2𝜋|ℎ(𝑧)|²𝑑𝜃≤1+3𝑟²1−𝑟². (10)

2. Some Properties of the Class 𝑘−𝒰𝒯_𝑚(𝛾)

In this section, we provide some of the interesting properties of class 𝑘−𝒰𝒯_𝑚(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 function 𝑓(𝑧)∈𝑘−𝒰𝒯_𝑚(𝛾) if and only if 𝑓^′(𝑧)=𝑓₁(𝑧)^{((𝑚/4)+(1/2))𝛾}𝑓₂(𝑧)^{((𝑚/4)−(1/2))𝛾}, (11) where 𝑓₁(𝑧) and 𝑓₂(𝑧) are close-to-convex functions.

Theorem 4. Let 𝑓∈𝑘−𝒰𝒯_𝑚(𝜌,𝛾) in 𝐸. Then, 𝑓∈𝐶 for |𝑧|<𝑟₀, where 𝑟₀=2(1+𝑘)𝑚𝛾+2𝑘+2+𝑚𝛾+2𝑘+2²().−41+𝑘2𝛾−𝑘−1 (12) This result is sharp.

Proof. We can write 𝑓^′(𝑧)=𝑔^′(𝑧)ℎ(𝑧),𝑔(𝑧)∈𝑘−𝒰𝒱_𝑚𝛾,ℎ(𝑧)∈𝑃. (13)
Using Lemma 1(ii), we get 𝑓^′(𝑧)=𝑠₁(𝑧)/𝑧^{((𝑚/4)+(1/2))}𝑠₂(𝑧)/𝑧^{((𝑚/4)−(1/2))𝛾/(1+𝑘)}ℎ(𝑧), (14) where 𝑠₁ and 𝑠₂ are starlike functions. Logarithmic differentiation of (14) gives us 𝑧𝑓^′′(𝑧)𝑓^′(𝑧)=𝛾𝑚1+𝑘−1+4+12𝑧𝑠^′₁(𝑧)𝑠₁(𝑧)−𝑚4−12𝑧𝑠^′₂(𝑧)𝑠₂(𝑧)+𝑧ℎ^′(𝑧)ℎ(𝑧). (15)
This implies that 1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)=1+𝑘−𝛾+𝛾1+𝑘𝑚1+𝑘4+12𝑧𝑠^′₁(𝑧)𝑠₁(𝑧)−𝑚4−12𝑧𝑠^′₂(𝑧)𝑠₂(𝑧)+𝑧ℎ^′(𝑧)ℎ(𝑧). (16) Now using distortion results for the class 𝑃, we have Re1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)≥1+𝑘−𝛾+𝛾1+𝑘𝑚1+𝑘4+121−𝑟−𝑚1+𝑟4−121+𝑟−1−𝑟2𝑟1−𝑟²=1+𝑘−𝛾1−𝑟²+𝛾1+𝑟²−𝑚𝑟−2𝑟(1+𝑘)(1+𝑘)1−𝑟². (17) 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−𝒰𝒯_𝑚(𝛾), given by 𝑓^′₀(𝑧)=(1+𝑧)^{(𝛾/(𝑘+1))(((𝑚/2)−1)+1)}(1−𝑧)^{(𝛾/(𝑘+1))(((𝑚/2)+1)+1)},𝑧∈𝐸. (18)
We note the following interesting special cases:(i)For 𝛾=1, we have the radius of convexity for class 𝑘−𝒰𝒯_𝑚.(ii)For 𝛾=1  and  𝑘=0, we have the radius of convexity for class 𝒯_𝑚, proved by Noor [10].(iii)For 𝛾=1, 𝑘=0  and  𝑚=2, we have radius of convexity for close-to-convex functions which is well known.

Theorem 5. Let 𝑓∈𝑘−𝒰𝒯_𝑚(𝛾) with 𝑘≥0, 𝑚≥2, and ((𝑚+2)/(1+𝑘))Re𝛾>1. Then, 𝐿_𝑟𝑓1≤𝐴𝑘,𝛾,𝑚1−𝑟^{(1/2)((𝑚+2)/(1+𝑘))Re𝛾}. (19) The exponent (1/2)((𝑚+2)/(1+𝑘))Re𝛾 is sharp.

Proof. Let 𝑓∈𝑘−𝒰𝒯_𝑚(𝜌,𝛾). Then, there exists 𝑔(𝑧)∈𝑘−𝒰𝒱_𝑚(𝛾) such that 𝑓^′(𝑧)=𝑔^′(𝑧)ℎ(𝑧),ℎ∈𝑃. (20) From the definition of 𝑘−𝒰𝒱_𝑚(𝛾), one can deduce that 𝑔(𝑧)∈𝑘−𝒰𝒱_𝑚(1) implies that 𝑔(𝑧)∈𝒱_𝑚(𝑘/(𝑘+1)).
Now using (20), Lemma 1(ii), and distortion theorems for starlike functions, we have 𝐿_𝑟𝑓=₀^2𝜋𝑧𝑔^′(𝑧)ℎ(𝑧𝑔(𝑧))𝑑𝜃,∈𝑘−𝒰𝒱_𝑚𝛾,ℎ(𝑧)∈𝑃=₀^2𝜋𝑧𝑠₁(𝑧)/𝑧^{((𝑚/4)+(1/2))(1/(1+𝑘))𝛾}𝑠₂(𝑧)/𝑧^{((𝑚/4)−(1/2))(1/(1+𝑘))𝛾}ℎ(𝑧=|)|𝑑𝜃_02𝜋𝑧^{1−𝛾(1/(𝑘+1))}𝑠₁(𝑧)^{((𝑚/4)+(1/2))(1/(1+𝑘))𝛾}𝑠₂(𝑧)^{((𝑚/4)−(1/2))(1/(1+𝑘))𝛾}×|ℎ(𝑧≤2)|𝑑𝜃_{((𝑚/2)−1)(1/(𝑘+1))Re𝛾}𝑟^{((𝑚/4)+(1/2))(1/(𝑘+1))Re𝛾−1}×₀^2𝜋𝑠₁(𝑧)^{(1/4)((𝑚+2)/(1+𝑘))Re𝛾}|ℎ(𝑧)|𝑑𝜃. (21) By using Hölder's inequality, this gives 𝐿_𝑟𝑓≤2^{((𝑚/2)−1)(1/(𝑘+1))Re𝛾}𝑟^{((𝑚/4)+(1/2))(1/(𝑘+1))Re𝛾−1}×12𝜋₀^2𝜋𝑠₁(𝑧)^{(1/2)((𝑚+2)/(1+𝑘))Re𝛾}𝑑𝜃^1/2×12𝜋₀^2𝜋|ℎ(𝑧)|²𝑑𝜃^1/2. (22) Since ((𝑚+2)/(1+𝑘))Re𝛾>1, therefore subordination for starlike functions and Lemma 2 give us 𝐿_𝑟𝑓1≤𝐴𝑘,𝛾,𝑚1−𝑟^{(1/2)((𝑚+2)/(1+𝑘))Re𝛾}. (23) The function 𝐹₀(𝑧)∈𝑘−𝒰𝒯_𝑚(𝛾) is defined by 𝐹^′₀(𝑧)=𝐺^′₀(𝑧)ℎ₀(𝑧), (24) where 𝐺^′₀(𝑧)=(1+𝑧)^{((𝑚/2)−1)(1/(1+𝑘))𝛾}(1−𝑧)^{((𝑚/2)+1)(1/(1+𝑘))𝛾},ℎ₀(𝑧)=1+𝑧1−𝑧 (25) shows that the exponent is sharp.

Some special choices in the above theorem give us the following interesting results.

Corollary 6. Let 𝑓∈𝑘−𝒰𝒯₂(1). Then 𝐿_𝑟𝑓(𝑘)1≤𝐴1−𝑟^2/(1+𝑘). (26)

Corollary 7. Let 𝑓(𝑧)∈𝒯_𝑚. Then 𝐿_𝑟𝑓(𝑚)1≤𝐴1−𝑟^(𝑚/2)+1. (27)

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−𝒰𝒯_𝑚(𝛾) by varying different parameters.

Theorem 8. Let 𝑓∈𝑘−𝒰𝒯_𝑚(𝛾) with 𝑘≥0, 𝑚≥2 and ((𝑚+2)/(1+𝑘))Re𝛾>1. Then 𝑎_𝑛=𝑂(1)𝑛^{((𝑚/2)+1)(Re𝛾/(1+𝑘))−1},(𝑛⟶∞). (28) The exponent is sharp.

Proof. With 𝑧=𝑟𝑒^𝑖𝜃, Cauchy’s theorem gives us 𝑛𝑎_𝑛=12𝜋𝑟^𝑛₀^2𝜋𝑧𝑓^′(𝑧1)𝑑𝜃=2𝜋𝑟^𝑛𝐿_𝑟𝑓,𝑧=𝑟𝑒^𝑖𝜃. (29) 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. Let 𝑓∈𝑘−𝒰𝒯_𝑚(1), and let it be of the form (1). Then, for 𝑛>3,  𝑘≥2, one has 𝑎_𝑛=𝑂(1)𝑛^{((𝑚/2)+1)(1/(1+𝑘))−1}. (30) For 𝑘=0, in the above corollary, we have growth rate of coefficients problem for functions in class 𝒯_𝑚, and, for 𝑘=0, 𝑚=2 gives us the growth rate of coefficients for close-to-convex functions, which are well known.

Acknowledgments

The authors want to acknowledge worthy referees of this paper for their insightful comments which greatly improves the entire presentation of the paper. They would also like to thank Prof. Dr. Ehsan Ali, VC AWKUM, for providing research facilities.