The Fekete-Szegö Problem for 𝑝-Valently Janowski Starlike and Convex Functions

Abstract

For 𝑝-valently Janowski starlike and convex functions defined by applying subordination for the generalized Janowski function, the sharp upper bounds of a functional |𝑎_𝑝+2−𝜇𝑎²_𝑝+1| related to the Fekete-Szegö problem are given.

1. Introduction

Let 𝒜_𝑝 denote the family of functions 𝑓(𝑧) normalized by 𝑓(𝑧)=𝑧^𝑝+^∞_𝑛=𝑝+1𝑎_𝑛𝑧^𝑛𝑝=1,2,3,…, (1.1) which are analytic in the open unit disk 𝕌={𝑧∈ℂ∶|𝑧|<1}. Furtheremore, let 𝒲 be the class of functions 𝑤(𝑧) of the form 𝑤(𝑧)=^∞_𝑘=1𝑤_𝑘𝑧^𝑘, (1.2) which are analytic and satisfy |𝑤(𝑧)|<1 in 𝕌. Then, a function 𝑤(𝑧)∈𝒲 is called the Schwarz function. If 𝑓(𝑧)∈𝒜_𝑝 satisfies the following condition 1Re1+𝑏𝑧𝑓^′(𝑧)𝑓(𝑧)−𝑝>0(𝑧∈𝕌), (1.3) for some complex number 𝑏(𝑏≠0), then 𝑓(𝑧) is said to be 𝑝-valently starlike function of complex order 𝑏. We denote by 𝒮^∗_𝑏(𝑝) the subclass of 𝒜_𝑝 consisting of all functions 𝑓(𝑧) which are 𝑝-valently starlike functions of complex order 𝑏. Similarly, we say that 𝑓(𝑧) is a member of the class 𝒦_𝑏(𝑝) of 𝑝-valently convex functions of complex order 𝑏 in 𝕌 if 𝑓(𝑧)∈𝒜_𝑝 satisfies 1Re1+𝑏𝑧𝑓^′′(𝑧)𝑓^′(𝑧)−𝑝−1>0(𝑧∈𝕌), (1.4) for some complex number 𝑏(𝑏≠0).

Next, let 𝐹(𝑧)=𝑧𝑓^′(𝑧)/𝑓(𝑧)=𝑢+𝑖𝑣 and 𝑏=𝜌𝑒^𝑖𝜑(𝜌>0,0≦𝜑<2𝜋). Then, the condition of the definition of 𝒮^∗_𝑏(𝑝) is equivalent to 1Re1+𝑏𝑧𝑓^′(𝑧)𝑓(𝑧)−𝑝=1+cos𝜑𝜌𝑢−𝑝+sin𝜑𝜌𝑣>0. (1.5)

We denote by 𝑑(𝑙₁,𝑝) the distance between the boundary line 𝑙₁∶(cos𝜑)𝑢+(sin𝜑)𝑣+𝜌−𝑝cos𝜑=0 of the half plane satisfying the condition (1.5) and the point 𝐹(0)=𝑝. A simple computation gives us that 𝑑𝑙₁,𝑝=cos𝜑×𝑝+sin𝜑×0+𝜌−𝑝cos𝜑cos²𝜑+sin²𝜑=𝜌, (1.6) that is, that 𝑑(𝑙₁,𝑝) is always equal to |𝑏|=𝜌 regardless of 𝜑. Thus, if we consider the circle 𝐶₁ with center at 𝑝 and radius 𝜌, then we can know the definition of 𝒮^∗_𝑏(𝑝) means that 𝐹(𝕌) is covered by the half plane separated by a tangent line of 𝐶₁ and containing 𝐶₁. For 𝑝=1, the same things are discussed by Hayami and Owa [3].

Then, we introduce the following function: 𝑝(𝑧)=1+𝐴𝑧(),1+𝐵𝑧−1≦𝐵<𝐴≦1 (1.7) which has been investigated by Janowski [4]. Therefore, the function 𝑝(𝑧) given by (1.7) is said to be the Janowski function. Furthermore, as a generalization of the Janowski function, Kuroki et al. [6] have investigated the Janowski function for some complex parameters 𝐴 and 𝐵 which satisfy one of the following conditions: (i)𝐴≠𝐵,|𝐵|<1,|𝐴|≦1,Re1−𝐴𝐵≧||;(𝐴−𝐵ii)𝐴≠𝐵,|𝐵|=1,|𝐴|≦1,1−𝐴𝐵>0. (1.8) Here, we note that the Janowski function generalized by the conditions (1.8) is analytic and univalent in 𝕌, and satisfies Re(𝑝(𝑧))>0(𝑧∈𝕌). Moreover, Kuroki and Owa [5] discussed the fact that the condition |𝐴|≦1 can be omitted from among the conditions in (1.8)-(i) as the conditions for 𝐴 and 𝐵 to satisfy Re(𝑝(𝑧))>0. In the present paper, we consider the more general Janowski function 𝑝(𝑧) as follows: 𝑝(𝑧)=𝑝+𝐴𝑧,1+𝐵𝑧𝑝=1,2,3,… (1.9) for some complex parameter 𝐴 and some real parameter 𝐵(𝐴≠𝑝𝐵,−1≦𝐵≦0). Then, we don't need to discuss the other cases because for the function: 𝑞(𝑧)=𝑝+𝐴₁𝑧1+𝐵₁𝑧𝐴₁,𝐵₁∈ℂ,𝐴₁≠𝑝𝐵₁,𝐵₁≦1, (1.10) letting 𝐵₁=|𝐵₁|𝑒^𝑖𝜃 and replacing 𝑧 by −𝑒^−𝑖𝜃𝑧 in (1.10), we see that 𝑝(𝑧)=𝑞−𝑒^−𝑖𝜃𝑧=𝑝−𝐴₁𝑒^−𝑖𝜃𝑧1−𝐵₁𝑧≡𝑝+𝐴𝑧1+𝐵𝑧𝐴=−𝐴₁𝑒^−𝑖𝜃,𝐵=−𝐵₁, (1.11) maps 𝕌 onto the same circular domain as 𝑞(𝕌).

Remark 1.1. For the case 𝐵=−1 in (1.9), we know that 𝑝(𝑧) maps 𝕌 onto the following half plane: Re𝑝+𝐴𝑝(𝑧)>𝑝²−|𝐴|²2, (1.12) and for the case −1<𝐵≦0 in (1.9), 𝑝(𝑧) maps 𝕌 onto the circular domain 𝑝(𝑧)−𝑝+𝐴𝐵1−𝐵²<𝐴+𝑝𝐵1−𝐵². (1.13)

Let 𝑝(𝑧) and 𝑞(𝑧) be analytic in 𝕌. Then, we say that the function 𝑝(𝑧) is subordinate to 𝑞(𝑧) in 𝕌, written by 𝑝(𝑧)≺𝑞(𝑧)(𝑧∈𝕌), (1.14) if there exists a function 𝑤(𝑧)∈𝒲 such that 𝑝(𝑧)=𝑞(𝑤(𝑧))(𝑧∈𝕌). In particular, if 𝑞(𝑧) is univalent in 𝕌, then 𝑝(𝑧)≺𝑞(𝑧) if and only if 𝑝(0)=𝑞(0),𝑝(𝕌)⊂𝑞(𝕌). (1.15)

We next define the subclasses of 𝒜_𝑝 by applying the subordination as follows: 𝒮^∗_𝑝(𝐴,𝐵)=𝑓(𝑧)∈𝒜_𝑝∶𝑧𝑓^′(𝑧)𝑓(𝑧)≺𝑝+𝐴𝑧(,𝒦1+𝐵𝑧𝑧∈𝕌)_𝑝(𝐴,𝐵)=𝑓(𝑧)∈𝒜_𝑝∶1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)≺𝑝+𝐴𝑧(,1+𝐵𝑧𝑧∈𝕌) (1.16) where 𝐴≠𝑝𝐵, −1≦𝐵≦0. We immediately know that 𝑓(𝑧)∈𝒦_𝑝(𝐴,𝐵)iff𝑧𝑓^′(𝑧)𝑝∈𝒮^∗_𝑝(𝐴,𝐵). (1.17)

Then, we have the next theorem.

Theorem 1.2. If 𝑓(𝑧)∈𝒮^∗_𝑝(𝐴,𝐵)(−1<𝐵≦0), then 𝑓(𝑧)∈𝒮^∗_𝑏(𝑝), where 𝐵(𝐴)(𝐴)𝑏=−𝑝𝐵+Recos𝜑+𝐵Imsin𝜑+𝐴−𝑝𝐵1−𝐵²𝑒^𝑖𝜑0≦𝜑<2𝜋. (1.18) Espesially, 𝑓(𝑧)∈𝒮^∗_𝑝(𝐴,−1) if and only if 𝑓(𝑧)∈𝒮^∗_𝑏(𝑝) where 𝑏=(𝑝+𝐴)/2.

Proof. Supposing that 𝑧𝑓^′(𝑧)/𝑓(𝑧)≺(𝑝+𝐴𝑧)/(1−𝑧), it follows from Remark 1.1 that Re𝑝+𝐴𝑧𝑓^′(𝑧)𝑓(𝑧)>𝑝²−|𝐴|²2, (1.19) that is, that Re2𝑝+𝐴𝑧𝑓′(𝑧)𝑓(𝑧)>Re2𝑝𝑝+𝐴−𝑝+𝐴². (1.20) This means that 2Re𝑝+𝐴𝑝+𝐴²𝑧𝑓^′(𝑧)𝑓(𝑧)−𝑝>−1, (1.21) which implies that 1Re(1/2)𝑝+𝐴𝑧𝑓^′(𝑧)𝑓(𝑧)−𝑝>−1. (1.22) Therefore, 𝑓(𝑧)∈𝒮^∗_𝑏, where 𝑏=(𝑝+𝐴)/2. The converse is also completed.
Next, for the case −1<𝐵≦0, by the definition of the class 𝒮^∗_𝑏(𝐴,𝐵), if a tangent line 𝑙₂ of the circle 𝐶₂ containing the point 𝑝 is parallel to the straight line 𝐿∶(cos𝜃)𝑢+(sin𝜃)𝑣=0(−𝜋≦^∃𝜃<𝜋), and the image 𝐹(𝕌) by 𝐹(𝑧)=𝑧𝑓^′(𝑧)/𝑓(𝑧) is covered by the circle 𝐶₂, then there exists a non-zero complex number 𝑏 with arg(𝑏)=𝜃+𝜋 and |𝑏|=𝑑(𝑙₂,𝑝) such that 𝑓(𝑧)∈𝒮^∗_𝑏(𝑝), where 𝑑(𝑙₂,𝑝) is the distance between the tangent line 𝑙₂ and the point 𝑝. Now, for the function 𝑝(𝑧)=(𝑝+𝐴𝑧)/(1+𝐵𝑧)(𝐴≠𝑝𝐵,−1<𝐵≦0), the image 𝑝(𝕌) is equivalent to 𝐶₂=𝜔∈ℂ∶𝜔−𝑝−𝐴𝐵1−𝐵²<𝐴−𝑝𝐵1−𝐵², (1.23) and the point 𝜉 on 𝜕𝐶₂={𝜔∈ℂ∶|𝜔−(𝑝−𝐴𝐵)/(1−𝐵²)|=|𝐴−𝑝𝐵|/(1−𝐵²)} can be written by 𝜉∶=𝜉(𝜃)=𝐴−𝑝𝐵1−𝐵²𝑒^𝑖𝜃+𝑝−𝐴𝐵1−𝐵²−𝜋≦^∃𝜃<𝜋. (1.24) Further, the tangent line 𝑙₂ of the circle 𝐶₂ through each point 𝜉(𝜃) is parallel to the straight line 𝐿∶(cos𝜃)𝑢+(sin𝜃)𝑣=0. Namely, 𝑙₂ can be represented by 𝑙₂∶()(𝐴)cos𝜃𝑢−𝐴−𝑝𝐵cos𝜃+𝑝−𝐵Re1−𝐵²+()(𝐴)sin𝜃𝑣−𝐴−𝑝𝐵sin𝜃−𝐵Im1−𝐵²=0, (1.25) which implies that 𝑙₂∶()()+(𝐴(𝐴)cos𝜃𝑢+sin𝜃𝑣−𝐴−𝑝𝐵𝑝−𝐵Re)cos𝜃−𝐵Imsin𝜃1−𝐵²=0. (1.26) Then, we see that the distance 𝑑(𝑙₂,𝑝) between the point 𝑝 and the above tangent line 𝑙₂ of the circle 𝐶₂ is +(𝐴(𝐴)cos𝜃×𝑝+sin𝜃×0−𝐴−𝑝𝐵𝑝−𝐵Re)cos𝜃−𝐵Imsin𝜃1−𝐵²=−𝐵−𝑝𝐵+Re(𝐴)cos𝜃−𝐵Im(𝐴)sin𝜃+𝐴−𝑝𝐵1−𝐵². (1.27) Therefore, if the subordination 𝑧𝑓^′(𝑧)𝑓(𝑧)≺𝑝+𝐴𝑧1+𝐵𝑧𝐴≠𝑝𝐵,−1<𝐵≦0 (1.28) holds true, then 𝑓(𝑧)∈𝒮^∗_𝑏 where (𝐴)(𝐴)𝑏=−𝐵−𝑝𝐵+Recos𝜃−𝐵Imsin𝜃+𝐴−𝑝𝐵1−𝐵²𝑒^{𝑖(𝜃+𝜋)}. (1.29) Finally, setting 𝜑=𝜃+𝜋(0≦𝜑<2𝜋), the proof of the theorem is completed.

Noonan and Thomas [8, 9] have stated the 𝑞th Hankel determinant as 𝐻_𝑞(𝑛)𝑎=det^𝑛𝑎_𝑛+1⋯𝑎_{𝑛+𝑞−1}𝑎_𝑛+1𝑎_𝑛+2⋯𝑎_𝑛+𝑞𝑎⋮⋮⋱⋮_{𝑛+𝑞−1}𝑎_𝑛+𝑞⋯𝑎_{𝑛+2𝑞−2}𝑛,𝑞∈ℕ={1,2,3,…}. (1.30) This determinant is discussed by several authors with 𝑞=2. For example, we can know that the functional |𝐻₂(1)|=|𝑎₃−𝑎²₂| is known as the Fekete-Szegö problem, and they consider the further generalized functional |𝑎₃−𝜇𝑎²₂|, where 𝑎₁=1 and 𝜇 is some real number (see, [1]). The purpose of this investigation is to find the sharp upper bounds of the functional |𝑎_𝑝+2−𝜇𝑎²_𝑝+1| for functions 𝑓(𝑧)∈𝒮^∗_𝑝(𝐴,𝐵) or 𝒦_𝑝(𝐴,𝐵).

2. Preliminary Results

We need some lemmas to establish our results. Applying the Schwarz lemma or subordination principle.

Lemma 2.1. If a function 𝑤(𝑧)∈𝒲, then 𝑤₁≦1. (2.1) Equality is attained for 𝑤(𝑧)=𝑒^𝑖𝜃𝑧 for any 𝜃∈ℝ.

The following lemma is obtained by applying the Schwarz-Pick lemma (see, e.g., [7]).

Lemma 2.2. For any functions 𝑤(𝑧)∈𝒲, 𝑤₂≦1−𝑤₁² (2.2) holds true. Namely, this gives us the following representation: 𝑤₂=1−𝑤₁²𝜁, (2.3) for some 𝜁(|𝜁|≦1).

3. 𝑝-Valently Janowski Starlike Functions

Our first main result is contained in

Theorem 3.1. If 𝑓(𝑧)∈𝒮^∗_𝑝(𝐴,𝐵), then 𝑎_𝑝+2−𝜇𝑎²_𝑝+1≦𝐴−𝑝𝐵1−2𝜇𝐴−𝑝+1−2𝑝𝜇𝐵21−2𝜇𝐴−𝑝+1−2𝑝𝜇𝐵≧1,𝐴−𝑝𝐵21−2𝜇𝐴−𝑝+1−2𝑝𝜇𝐵≦1, (3.1) with equality for 𝑓(𝑧)=𝑧^𝑝(1+𝐵𝑧)^{(𝑝𝐵−𝐴)/𝐵}or𝑧^𝑝𝑒^𝐴𝑧()𝐵,𝑧𝐵=01−2𝜇𝐴−𝑝+1−2𝑝𝜇≧1_𝑝1+𝐵𝑧^{2(𝑝𝐵−𝐴)/2𝐵}or𝑧^𝑝𝑒^{(𝐴/2)𝑧2}(𝐵=0)1−2𝜇𝐴−𝑝+1−2𝑝𝜇𝐵≦1. (3.2)

Proof. Let 𝑓(𝑧)∈𝒮^∗_𝑝(𝐴,𝐵). Then, there exists the function 𝑤(𝑧)∈𝒲 such that 𝑧𝑓^′(𝑧)𝑓(𝑧)=𝑝+𝐴𝑤(𝑧)(𝑧),1+𝐵𝑤 (3.3) which means that 𝑛−𝑝𝑎_𝑛=^𝑛−1_𝑘=𝑝(𝐴−𝑘𝐵)𝑎_𝑘𝑤_𝑛−𝑘𝑛≧𝑝+1, (3.4) where 𝑎_𝑝=1. Thus, by the help of the relation in Lemma 2.2, we see that 𝑎_𝑝+2−𝜇𝑎²_𝑝+1=12𝐴−𝑝𝐵𝑤₂+𝐴−𝑝+1𝐵𝑤²₁−𝜇𝐴−𝑝𝐵²𝑤²₁=𝐴−𝑝𝐵21−𝑤²₁𝜁+𝐴−𝑝+1𝐵−2𝜇𝐴−𝑝𝐵𝑤²₁. (3.5) Then, by Lemma 2.1, supposing that 0≦𝑤₁≦1 without loss of generality, and applying the triangle inequality, it follows that 1−𝑤²₁𝜁+𝐴−𝑝+1𝐵−2𝜇𝐴−𝑝𝐵𝑤²₁≦1+𝐴−𝑝+1𝐵−2𝜇𝐴−𝑝𝐵−1𝑤²₁≦𝐴−𝑝+1𝐵−2𝜇𝐴−𝑝𝐵𝐴−𝑝+1𝐵−2𝜇𝐴−𝑝𝐵≧1;𝑤₁,1𝐵=1𝐴−𝑝+1−2𝜇𝐴−𝑝𝐵≦1;𝑤₁=0. (3.6)

Especially, taking 𝜇=(𝑝+1)/2𝑝 in Theorem 3.1, we obtain the following corollary.

Corollary 3.2. If 𝑓(𝑧)∈𝒮^∗_𝑝(𝐴,𝐵), then 𝑎_𝑝+2−𝑝+1𝑎2𝑝^2𝑝+1≦𝐴𝐴−𝑝𝐵|𝐴|,2𝑝≧𝑝𝐴−𝑝𝐵2|𝐴|≦𝑝, (3.7) with equality for 𝑓(𝑧)=𝑧^𝑝(1+𝐵𝑧)^{(𝑝𝐵−𝐴)/𝐵}or𝑧^𝑝𝑒^𝐴𝑧()|𝐴|,𝑧𝐵=0≧𝑝_𝑝1+𝐵𝑧^{2(𝑝𝐵−𝐴)/2𝐵}or𝑧^𝑝𝑒^{(𝐴/2)𝑧2}(𝐵=0)|𝐴|≦𝑝. (3.8)

Furthermore, putting 𝐴=𝑝−2𝛼 and 𝐵=−1 for some 𝛼(0≦𝛼<𝑝) in Theorem 3.1, we arrive at the following result by Hayami and Owa [2, Theorem  3].

Corollary 3.3. If 𝑓(𝑧)∈𝒮^∗_𝑝(𝛼), then 𝑎_𝑝+2−𝜇𝑎²_𝑝+1≦2𝜇1𝑝−𝛼𝑝−𝛼+1−4𝑝−𝛼𝜇≦2,1𝑝−𝛼2≦𝜇≦𝑝−𝛼+12,42𝑝−𝛼𝑝−𝛼𝑝−𝛼𝜇−𝑝−𝛼+1𝜇≧𝑝−𝛼+12,𝑝−𝛼 (3.9) with equality for 𝑓(𝑧)=𝑧(1−𝑧)^{2(𝑝−𝛼)}1𝜇≦2or𝜇≧𝑝−𝛼+12,𝑧𝑝−𝛼1−𝑧^2𝑝−𝛼12≦𝜇≦𝑝−𝛼+12.𝑝−𝛼 (3.10)

4. 𝑝-Valently Janowski Convex Functions

Similarly, we consider the functional |𝑎_𝑝+2−𝜇𝑎²_𝑝+1| for 𝑝-valently Janowski convex functions.

Theorem 4.1. If 𝑓(𝑧)∈𝒦_𝑝(𝐴,𝐵), then 𝑎_𝑝+2−𝜇𝑎²_𝑝+1≦𝑝𝐴−𝑝𝐵𝑝+1²−2𝑝𝑝+2𝜇𝐴−𝑝+1³−2𝑝²𝑝+2𝜇𝐵2𝑝+1²,𝑝+2𝑝+1²−2𝑝𝑝+2𝜇𝐴−𝑝+1³−2𝑝²𝑝+2𝜇𝐵≧𝑝+1²,𝑝𝐴−𝑝𝐵2,𝑝+2𝑝+1²−2p𝑝+2𝜇𝐴−𝑝+1³−2𝑝²𝑝+2𝜇𝐵≦𝑝+1², (4.1) with equality for 𝑓(𝑧)=𝑧^𝑝₂𝐹₁𝐴𝑝,𝑝−𝐵;𝑝+1;−𝐵𝑧or𝑧^𝑝₁𝐹₁𝑝,𝑝+1;𝐴𝑧(𝐵=0),𝑝+1²−2𝑝𝑝+2𝜇𝐴−𝑝+1³−2𝑝²𝑝+2𝜇𝐵≧𝑝+1²,𝑧^𝑝₂𝐹₁𝑝2,𝑝𝐵−𝐴𝑝2𝐵;1+2;−𝐵𝑧²or𝑧^𝑝₁𝐹₁𝑝2𝑝,1+2;𝐴2𝑧²(𝐵=0),𝑝+1²−2𝑝𝑝+2𝜇𝐴−𝑝+1³−2𝑝²𝑝+2𝜇𝐵≦𝑝+1², (4.2) where ₂𝐹₁(𝑎,𝑏;𝑐;𝑧) represents the ordinary hypergeometric function and ₁𝐹₁(𝑎,𝑏;𝑧) represents the confluent hypergeometric function.

Proof. By the help of the relation (1.17) and Theorem 3.1, if 𝑓(𝑧)∈𝒦_𝑝(𝐴,𝐵), then 𝑝+2𝑝𝑎_𝑝+2−𝜇𝑝+1²𝑝²𝑎²_𝑝+1=𝑝+2𝑝𝑎_𝑝+2−𝑝+1²𝑝𝑝+2𝜇𝑎²_𝑝+1,≦𝐶𝜇, (4.3) where 𝐶(𝜇) is one of the values in Theorem 3.1. Then, dividing the both sides by (𝑝+2)/𝑝 and replacing ((𝑝+1)²/𝑝(𝑝+2))𝜇 by 𝜇, we obtain the theorem.

Now, letting 𝜇=(𝑝+1)³/2𝑝²(𝑝+2) in Theorem 4.1, we have the following corollary.

Corollary 4.2. If 𝑓(𝑧)∈𝒦_𝑝(𝐴,𝐵), then 𝑎_𝑝+2−𝑝+1³2𝑝²𝑎𝑝+2^2𝑝+1≦𝐴𝐴−𝑝𝐵2|𝐴|,𝑝𝑝+2≧𝑝𝐴−𝑝𝐵2|𝐴|,𝑝+2≦𝑝 (4.4) wiht equality for 𝑓(𝑧)=𝑧^𝑝₂𝐹₁𝐴𝑝,𝑝−𝐵;𝑝+1;−𝐵𝑧or𝑧^𝑝₁𝐹₁)𝐴|,𝑧𝑝,𝑝+1;𝐴𝑧(𝐵=0|≧𝑝_𝑝2𝐹₁𝑝2,𝑝𝐵−𝐴𝑝2𝐵;1+2;−𝐵𝑧²or𝑧^𝑝₁𝐹₁𝑝2𝑝,1+2;𝐴2𝑧²(𝐵=0)|𝐴|≦𝑝, (4.5) where ₂𝐹₁(𝑎,𝑏;𝑐;𝑧) represents the ordinary hypergeometric function and ₁𝐹₁(𝑎,𝑏;𝑧) represents the confluent hypergeometric function.

Moreover, we suppose that 𝐴=𝑝−2𝛼 and 𝐵=−1 for some 𝛼(0≦𝛼<𝑝). Then, we arrive at the result by Hayami and Owa [2, Theorem  4].

Corollary 4.3. If 𝑓(𝑧)∈𝒦_𝑝(𝛼), then 𝑎_𝑝+2−𝜇𝑎²_𝑝+1≦𝑝𝑝−𝛼𝑝+1²2𝑝−𝛼+1−4𝑝𝑝+2𝑝−𝛼𝜇𝑝+1²𝑝+2𝜇≦𝑝+1²,𝑝2𝑝𝑝+2𝑝−𝛼𝑝+2𝑝+1²2𝑝𝑝+2≦𝜇≦𝑝+1²𝑝−𝛼+1,𝑝2𝑝𝑝+2𝑝−𝛼𝑝−𝛼4𝑝𝑝+2𝑝−𝛼𝜇−𝑝+1²2𝑝−𝛼+1𝑝+1²𝑝+2𝜇≧𝑝+1²𝑝−𝛼+1,2𝑝𝑝+2𝑝−𝛼 (4.6) with equality for 𝑓(𝑧)=𝑧^𝑝₂𝐹₁𝑝,2𝑝−𝛼;𝑝+1;𝑧𝜇≦𝑝+1²2𝑝𝑝+2or𝜇≧𝑝+1²𝑝−𝛼+1,𝑧2𝑝𝑝+2𝑝−𝛼_𝑝2𝐹₁𝑝2𝑝,𝑝−𝛼;1+2;𝑧²𝑝+1²2𝑝𝑝+2≦𝜇≦𝑝+1²𝑝−𝛼+1,2𝑝𝑝+2𝑝−𝛼 (4.7) where ₂𝐹₁(𝑎,𝑏;𝑐;𝑧) represents the ordinary hypergeometric function.