On a Subclass of Meromorphic Functions Defined by Hilbert Space Operator

Abstract

In this paper, we define a new operator on the class of meromorphic functions and define a subclass using Hilbert space operator. Coefficient estimate, distortion bounds, extreme points, radii of starlikeness, and convexity are obtained.

1. Introduction

Let Σ denote the class of meromorphic functions 𝑓(𝑧)=1𝑧+^∞_𝑛=1𝑎_𝑛𝑧^𝑛 (1) defined on 𝑈={𝑧∈𝐶∶0<|𝑧|<1}. For 𝑓∈Σ given by (1) and 𝑔∈Σ given by 𝑔(𝑧)=1𝑧+^∞_𝑛=1𝑏_𝑛𝑧^𝑛, (2) the Hadamard product (or convolution) [1] of 𝑓 and 𝑔 is defined by (𝑧)=1𝑓∗𝑔𝑧+^∞_𝑛=1𝑎_𝑛𝑏_𝑛𝑧^𝑛,𝑧∈𝑈. (3) Many subclasses of meromorphic functions have been defined and studied in the past. In particular, the subclasses 𝑀^𝛼1_𝑙,𝑚(𝜆,𝛼),  0≤𝛼<1 and 0≤𝜆<1  [2], and 𝑊(𝛼,𝛽),0≤𝛼<1 and 0<𝛽≤1  [3], are considered by researchers. Let 𝐻 be a complex Hilbert space and 𝐿(𝐻) denote the algebra of all bounded linear operators on 𝐻. For a complex-valued function 𝑓 analytic in a domain 𝐸 of the complex plane containing the spectrum 𝜎(𝑇) of the bounded linear operator 𝑇, let 𝑓(𝑇) denote the operator on 𝐻 defined by the Riesz-Dunford integral [4] 𝑓(𝑇)=12𝜋𝑖_𝐶(𝑧𝐼−𝑇)⁻¹𝑓(𝑧)𝑑𝑧, (4) where 𝐼 is the identity operator on 𝐻 and 𝐶 is a positively oriented simple closed rectifiable closed contour containing the spectrum 𝜎(𝑇) in the interior domain [5]. The operator 𝑓(𝑇) can also be defined by the following series: 𝑓(𝑇)=^∞_𝑛=0𝑓^(𝑛)(0)𝑇𝑛!^𝑛 (5) which converges in the norm topology. In this paper, we introduce a subclass 𝑀_𝑃(𝛼,𝜆,𝑇) of Σ defined using Hilbert space operator and prove a necessary and sufficient condition for the function to belong to this class, the distortion theorem, radius of starlikeness, and convexity. In [6], Atshan and Buti had defined an operator acting on analytic functions in terms of a definite integral. We modify their operator for meromorphic functions as follows.

Lemma 1. For 𝑓∈Σ given by (1), 0≤𝜇≤1, and 0≤𝜃≤1, if the operator 𝑆^𝜃_𝜇∶Σ→Σ is defined by 𝑆^𝜃_𝜇𝑓(𝑧)=11−𝜇^𝜃+1Γ(𝜃+1)^∞0𝑡^𝜃+1𝑒^{−(𝑡/(1−𝜇))}𝑓(𝑧𝑡)𝑑𝑡, (6) then 𝑆^𝜃_𝜇𝑓(𝑧)=1𝑧+^∞_𝑛=1𝐿𝑛,𝜃,𝜇𝑎_𝑛𝑧^𝑛, (7) where 𝐿(𝑛,𝜃,𝜇)=(1−𝜇)^𝑛+1Γ(𝑛+𝜃+2)/Γ(𝜃+1).

Denote by Σ_𝑃 the class of all functions 𝑓∈Σ with 𝑎_𝑛≥0.

Definition 2. For 0≤𝜆<1, 0≤𝛼<1, a function 𝑓∈Σ_𝑃 is in the class 𝑀_𝑃(𝛼,𝜆,𝑇) if 𝑇𝑆^𝜃_𝜇𝑓(𝑇)^′−(𝜆−1)𝑆^𝜃_𝜇𝑓(𝑇)+𝜆𝑇𝑆^𝜃_𝜇𝑓(𝑇)^′<𝑇𝑆^𝜃_𝜇𝑓(𝑇)^′+(1−2𝛼)(𝜆−1)𝑆^𝜃_𝜇𝑓(𝑇)+𝜆𝑇𝑆^𝜃_𝜇𝑓(𝑇)^′ (8) for all operators 𝑇 with ‖𝑇‖<1 and 𝑇≠0,  0 being the zero operator on 𝐻.

2. Coefficient Bounds

Theorem 3. A meromorphic function 𝑓∈Σ_𝑃 given by (1) is in the class 𝑀_𝑃(𝛼,𝜆,𝑇) if and only if ^∞_𝑛=1[𝑛+𝛼−𝛼𝜆(𝑛+1)]𝐿𝑛,𝜃,𝜇𝑎_𝑛≤1−𝛼. (9) The result is sharp for 𝑓(𝑧)=1/𝑧+((1−𝛼)/[𝑛+𝛼−𝛼𝜆(𝑛+1)]𝐿(𝑛,𝜃,𝜇))𝑧^𝑛.

Proof. Let 𝑓(𝑇)=𝑇⁻¹+∑^∞_𝑛=1𝑎_𝑛𝑇^𝑛. Assume that (9) holds. Then, 𝑇𝑆^𝜃_𝜇𝑓(𝑇)^′−(𝜆−1)𝑆^𝜃_𝜇𝑓(𝑇)+𝜆𝑇𝑆^𝜃_𝜇𝑓(𝑇)^′−𝑇𝑆^𝜃_𝜇𝑓(𝑇)^′+(1−2𝛼)(𝜆−1)𝑆^𝜃_𝜇𝑓(𝑇)+𝜆𝑇𝑆^𝜃_𝜇𝑓(𝑇)^′=^∞_𝑛=1[(𝑛+1)−𝜆(𝑛+1)]𝐿𝑛,𝜇,𝜃𝑎_𝑛𝑇^𝑛−2𝑇⁻¹()−1−𝛼_∞𝑛=1[()()𝑛]𝑎𝑛+1−2𝛼)(𝜆−1+𝜆1−2𝛼×𝐿𝑛,𝜇,𝜃_𝑛𝑇^𝑛≤^∞_𝑛=1[𝑛+𝛼−𝛼𝜆(𝑛+1)]𝐿𝑛,𝜃,𝜇𝑎_𝑛−()(9),1−𝛼≤0,by (10) and hence 𝑓∈𝑀_𝑃(𝛼,𝜆,𝑇).
Conversely, let 𝑇𝑆^𝜃_𝜇𝑓(𝑇)^′−(𝜆−1)𝑆^𝜃_𝜇𝑓(𝑇)+𝜆𝑇𝑆^𝜃_𝜇𝑓(𝑇)^′<𝑇𝑆^𝜃_𝜇𝑓(𝑇)^′+(1−2𝛼)(𝜆−1)𝑆^𝜃_𝜇𝑓(𝑇)+𝜆𝑇𝑆^𝜃_𝜇𝑓(𝑇)^′. (11) This implies that [(𝑛+1)(1−𝜆)]𝐿^∞_𝑛=1𝑛,𝜃,𝜇𝑎_𝑛𝑇^𝑛+1≤2(1−𝛼)−^∞_𝑛=1[()()𝑛]𝑎𝑛+1−2𝛼)(𝜆−1+𝜆1−2𝛼×𝐿𝑛,𝜃,𝜇_𝑛𝑇^𝑛+1. (12) Choose 𝑇=𝑒𝐼,  0<𝑒<1. Then, ∑^∞_𝑛=1[(𝑛+1)(1−𝜆)]𝐿(𝑛,𝜃,𝜇)𝑎_𝑛𝑒^𝑛/(2𝑒(1−𝛼)−∑^∞_𝑛=1[𝑛+(1−2𝛼)(𝜆−1)+𝜆(1−2𝛼)𝑛]𝐿(𝑛,𝜃⋅𝜇)𝑎_𝑛𝑒^𝑛)≤1. As 𝑒→1, we obtain (9).

Corollary 4. If 𝑓 of the form (1) is in 𝑀_𝑃(𝛼,𝜆,𝑇), then 𝑎_𝑛≤1−𝛼[(𝐿.𝑛+𝛼−𝛼𝜆𝑛+1)]𝑛,𝜃,𝜇 (13) The result is sharp for the function 𝑓(𝑧)=1/𝑧+((1−𝛼)/[𝑛+𝛼−𝛼𝜆(𝑛+1)]𝐿(𝑛,𝜃,𝜇))𝑧^𝑛.

Theorem 5. Let 𝑓₀(𝑧)=1/𝑧 and 𝑓_𝑛(𝑧)=1/𝑧+((1−𝛼)/[𝑛+𝛼−𝛼𝜆(𝑛+1)]𝐿(𝑛,𝜃,𝜇))𝑧^𝑛,  𝑛≥1,  0≤𝛼<1,  0≤𝜆<1. Then, 𝑓 is in the class 𝑀_𝑃(𝛼,𝜆,𝑇) if and only if it can be expressed in the form 𝑓(𝑧)=∑^∞_𝑛=0𝜇_𝑛𝑓_𝑛(𝑧), where 𝜇_𝑛≥0(𝑛=0,1,2,…) and ∑^∞_𝑛=0𝜇_𝑛=1.

Proof. Suppose that 𝑓(𝑧)=∑^∞_𝑛=0𝜇_𝑛𝑓_𝑛(𝑧)=1/𝑧+∑^∞_𝑛=1𝜇_𝑛((1−𝛼)/[𝑛+𝛼−𝛼𝜆(𝑛+1)]𝐿(𝑛,𝜃,𝜇))𝑧^𝑛; then, we have ^∞_𝑛=1[(𝐿×𝜇𝑛+𝛼−𝛼𝜆𝑛+1)]𝑛,𝜃,𝜇_𝑛(1−𝛼)[(𝐿=𝑛+𝛼−𝛼𝜆𝑛+1)]𝑛,𝜃,𝜇_∞𝑛=1𝜇_𝑛=1−𝜇₀<1. (14) By Theorem 3, 𝑓∈𝑀_𝑃(𝛼,𝜆,𝑇). Conversely, assume that 𝑓 is in the class 𝑀_𝑃(𝛼,𝜆,𝑇); then, by Corollary 4, 𝑎_𝑛≤1−𝛼[(𝐿.𝑛+𝛼−𝛼𝜆𝑛+1)]𝑛,𝜃,𝜇 (15) Set 𝜇_𝑛=([𝑛+𝛼−𝛼𝜆(𝑛+1)]𝐿(𝑛,𝜃,𝜇)/(1−𝛼))𝑎_𝑛, 𝑛=1,2,… and 𝜇₀=1−∑^∞_𝑛=1𝜇_𝑛. Then, 𝑓(𝑧)=∑^∞_𝑛=0𝜇_𝑛𝑓_𝑛(𝑧)∈𝑀_𝑃(𝛼,𝜆,𝑇).

3. Distortion Bounds

In this section, we obtain growth and distortion bounds for the class 𝑀_𝑃(𝛼,𝜆,𝑇).

Theorem 6. If 𝑓∈𝑀_𝑃(𝛼,𝜆,𝑇), then 𝑓(𝑇≥1)‖𝑇‖−1−𝛼(1+𝛼−2𝛼𝜆)1−𝜇²()‖𝑇‖,𝑓(𝑇≤1𝜃+1)(𝜃+2)‖𝑇‖+1−𝛼(1+𝛼−2𝛼𝜆)1−𝜇²()‖𝑇‖.𝜃+1)(𝜃+2 (16) The result is sharp for 𝑓(𝑧)=1/𝑧+((1−𝛼)/(1+𝛼−2𝛼𝜆)(1−𝜇)²(𝜃+1)(𝜃+2))𝑧.

Proof. By Theorem 3, (1−𝛼−2𝛼𝜆)𝐿1,𝜃,𝜇^∞_𝑛=2𝑎_𝑛≤^∞_𝑛=1[𝑛+𝛼−𝛼𝜆(𝑛+1)]𝐿𝑛,𝜃,𝜇𝑎_𝑛≤1−𝛼. (17) Therefore, ^∞_𝑛=1𝑎_𝑛≤1−𝛼()𝐿=1+𝛼−2𝛼𝜆1,𝜃,𝜇1−𝛼(1+𝛼−2𝛼𝜆)1−𝜇²().𝜃+1)(𝜃+2 (18) Also, if 𝑓(𝑇)=𝑇⁻¹+∑^∞_𝑛=1𝑎_𝑛𝑇^𝑛, then 1‖𝑇‖−^∞_𝑛=1𝑎_𝑛‖𝑇‖^𝑛≤𝑓(𝑇≤1)‖𝑇‖+^∞_𝑛=1𝑎_𝑛‖𝑇‖^𝑛. (19) Since ‖𝑇‖<1, the above inequality becomes 1‖𝑇‖−‖𝑇‖^∞_𝑛=1𝑎_𝑛≤𝑓(𝑇≤1)‖𝑇‖+‖𝑇‖^∞_𝑛=1𝑎_𝑛. (20) Using (18), we get the result.

Theorem 7. If 𝑓∈𝑀_𝑃(𝛼,𝜆,𝑇), then 𝑓^′(𝑇≥1)‖𝑇‖²−1−𝛼(1+𝛼−2𝛼𝜆)1−𝜇²(),𝑓𝜃+1)(𝜃+2_′(𝑇≤1)‖𝑇‖²+1−𝛼(1+𝛼−2𝛼𝜆)1−𝜇²().𝜃+1)(𝜃+2 (21) The result is sharp for 𝑓(𝑧)=1/𝑧+((1−𝛼)/(1+𝛼−2𝛼𝜆)(1−𝜇)²(𝜃+1)(𝜃+2))𝑧.

4. Radii Results

We now find the radius of meromorphically starlikeness and convexity for functions 𝑓 in the class 𝑀_𝑃(𝛼,𝜆,𝑇).

Theorem 8. Let 𝑓∈𝑀_𝑃(𝛼,𝜆,𝑇). Then, 𝑓 is meromorphically starlike of order 𝛿,  (0≤𝛿<1) in |𝑧|<𝑟₁, where 𝑟₁=inf_𝑛≥1(1−𝛿)(𝑛+𝛼−𝛼𝜆(𝑛+1))𝐿𝑛,𝜃,𝜇(1−𝛼)(𝑛+2−𝛿)^1/(𝑛+1),𝑛≥1. (22)

Proof. Let 𝑓(𝑇)=𝑇⁻¹+∑^∞_𝑛=1𝑎_𝑛𝑇^𝑛. Since 𝑓∈𝑀_𝑃(𝛼,𝜆,𝑇) is meromorphically starlike of order 𝛿, −𝑇𝑓^′(𝑇)𝑓(𝑇)−1≤1−𝛿. (23) Substituting for 𝑓, the above inequality becomes ^∞_𝑛=1𝑛+2−𝛿‖𝑇‖1−𝛿^𝑛+1𝑎_𝑛≤1. (24) By Theorem 3, ^∞_𝑛=1[𝑛+𝛼−𝛼𝜆(𝑛+1)]𝐿𝑛,𝜃,𝜇𝑎1−𝛼^𝑛≤1. (25) Thus, (24) will be true if 𝑛+2−𝛿‖𝑇‖1−𝛿^𝑛+1≤[𝑛+𝛼−𝛼𝜆(𝑛+1)]𝐿𝑛,𝜃,𝜇(),1−𝛼 (26) that is, ‖𝑇‖≤(1−𝛿)(𝑛+𝛼−𝛼𝜆(𝑛+1))𝐿𝑛,𝜃,𝜇(1−𝛼)(𝑛+2−𝛿)^1/(𝑛+1). (27)

Theorem 9. Let 𝑓∈𝑀_𝑃(𝛼,𝜆,𝑇). Then, 𝑓 is meromorphically convex of order 𝛿,(0≤𝛿<1) in |𝑧|<𝑟₂, where 𝑟₂=inf_𝑛≥1(1−𝛿)(𝑛+𝛼−𝛼𝜆(𝑛+1))𝐿𝑛,𝜃,𝜇(1−𝛼)𝑛(𝑛+2−𝛿)^1/(𝑛+1). (28)

Theorem 10. The class 𝑀_𝑃(𝛼,𝜆,𝑇) is closed under convex combination.

Proof. Let 𝑓(𝑧)=𝑇⁻¹+∑^∞_𝑛=0𝑎_𝑛𝑇^𝑛 and 𝑔(𝑧)=𝑇⁻¹+∑^∞_𝑛=0𝑏_𝑛𝑇^𝑛∈𝑀_𝑃(𝛼,𝜆,𝑇). Then, by Theorem 3, ^∞_𝑛=1[𝑛+𝛼−𝛼𝜆(𝑛+1)]𝐿𝑛,𝜃,𝜇𝑎_𝑛≤1−𝛼,_∞𝑛=1[𝑛+𝛼−𝛼𝜆(𝑛+1)]𝐿𝑛,𝜃,𝜇𝑏_𝑛≤1−𝛼. (29) Define ℎ(𝑇)=𝜆𝑓(𝑇)+(1−𝜆)𝑔(𝑇). Then, ℎ(𝑇)=𝑇⁻¹+∑^∞_𝑛=1[𝜆𝑎_𝑛+(1−𝜆)𝑏_𝑛]𝑇^𝑛. Now, ^∞_𝑛=1[𝑛+𝛼−𝛼𝜆(𝑛+1)]𝐿𝑛,𝜃,𝜇𝜆𝑎_𝑛+(1−𝜆)𝑏_𝑛=𝜆^∞_𝑛=1[𝑛+𝛼−𝛼𝜆(𝑛+1)]𝐿𝑛,𝜃,𝜇𝑎_𝑛+(1−𝜆)^∞_𝑛=1[𝑛+𝛼−𝛼𝜆(𝑛+1)]𝐿𝑛,𝜃,𝜇𝑏_𝑛≤𝜆(1−𝛼)+(1−𝜆)(1−𝛼)using(30)=1−𝛼. (30) Thus, ℎ∈𝑀_𝑃(𝛼,𝜆,𝑇).

5. Hadamard Product

Theorem 11. The Hadamard product 𝑓∗𝑔 of two functions 𝑓 and 𝑔 in 𝑀_𝑃(𝛼,𝜆,𝑇) belongs to the class 𝑀_𝑃(𝜉,𝜆,𝑇), where  𝜉=1−2(1−𝛼)²(1−𝜆)/(1−𝛼)²(1−2𝜆)+(1−𝛼−2𝛼𝜆)²(1−𝜇)²(𝜃+1)(𝜃+2).

Proof. It suffices to prove that 𝑛+𝜉−𝜉𝜆(^∞_𝑛=1𝑛+1)𝐿𝑛,𝜃,𝜇𝑎1−𝜉^𝑛𝑏_𝑛≤1, (31) where 𝜉=1−2(1−𝛼)²(×()1−𝜆)1−𝛼²(1−2𝜆)+(1−𝛼−2𝛼𝜆)²×1−𝜇²(𝜃+1)(𝜃+2)⁻¹. (32) Since 𝑓,𝑔∈𝑀_𝑃(𝛼,𝜆,𝑇)∑^∞_𝑛=1[𝑛+𝛼−𝛼𝜆(𝑛+1)]𝐿𝑛,𝜃,𝜇𝑎1−𝛼^𝑛∑≤1,_∞𝑛=1[𝑛+𝛼−𝛼𝜆(𝑛+1)]𝐿𝑛,𝜃,𝜇𝑏1−𝛼^𝑛≤1. (33) By Cauchy-Schwartz inequality, ∑^∞_𝑛=1[𝑛+𝛼−𝛼𝜆(𝑛+1)]𝐿𝑛,𝜃,𝜇1−𝛼𝑎_𝑛𝑏_𝑛≤1. (34) We have to find the largest 𝜉 such that 𝑛+𝜉−𝜉𝜆(𝑛+1)𝐿𝑛,𝜃,𝜇𝑎1−𝜉^𝑛𝑏_𝑛≤[𝑛+𝛼−𝛼𝜆(𝑛+1)]𝐿𝑛,𝜃,𝜇1−𝛼𝑎_𝑛𝑏_𝑛, (35) that is, 𝑎_𝑛𝑏_𝑛≤1−𝛼[(𝐿.𝑛+𝛼−𝛼𝜆𝑛+1)]𝑛,𝜃,𝜇 (36) It is enough to find the largest 𝜉 such that 1−𝛼[(𝐿≤((𝑛+𝛼−𝛼𝜆𝑛+1)]𝑛⋅𝜃,𝜇1−𝜉)[𝑛+𝛼−𝛼𝜆𝑛+1)](1−𝛼)𝑛+𝜉−𝜉𝜆(𝑛+1) (37) which yields 𝜉≤(𝑛+𝛼−𝛼𝜆(𝑛+1))²𝐿𝑛,𝜃,𝜇−(1−𝛼)²𝑛×(1−𝛼)²((+((1−𝜆𝑛+1))𝑛+𝛼−𝛼𝜆𝑛+1))²𝐿𝑛,𝜃,𝜇⁻¹. (38)
It follows from (38) that 𝜉≤1−(1−𝛼)²()×()𝑛+1)(1−𝜆1−𝛼²((+((1−𝜆𝑛+1))𝑛+𝛼−𝛼𝜆𝑛+1))²𝐿𝑛,𝜃,𝜇⁻¹. (39)
Let 𝛿(𝑛)=(1−𝛼)²()×()𝑛+1)(1−𝜆1−𝛼²((+((1−𝜆𝑛+1))𝑛+𝛼−𝛼𝜆𝑛+1))²𝐿𝑛,𝜃,𝜇⁻¹. (40)
Clearly 𝛿(𝑛) is an increasing function of 𝑛  (𝑛≥1). Letting 𝑛=1, we prove the assertion.