On a Subclass of Meromorphic Functions Defined by Hilbert Space ...

Hindawi Publishing Corporation Geometry Volume 2013, Article ID 671826, 4 pages http://dx.doi.org/10.1155/2013/671826

Research Article On a Subclass of Meromorphic Functions Defined by Hilbert Space Operator Thomas Rosy and S. Sunil Varma Department of Mathematics, Madras Christian College, Tambaram, Chennai, Tamil Nadu 600 059, India Correspondence should be addressed to Thomas Rosy; [email protected] Received 26 April 2013; Accepted 30 May 2013 Academic Editor: JinLin Liu Copyright © 2013 T. Rosy and S. S. Varma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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.

let 𝑓(𝑇) denote the operator on 𝐻 defined by the RieszDunford integral [4]

1. Introduction Let Σ denote the class of meromorphic functions

𝑓 (𝑇) =

∞

1 𝑓 (𝑧) = + ∑ 𝑎𝑛 𝑧𝑛 𝑧 𝑛=1

(1)

defined on 𝑈 = {𝑧 ∈ 𝐶 : 0 < |𝑧| < 1}. For 𝑓 ∈ Σ given by (1) and 𝑔 ∈ Σ given by

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(𝑛) (0) 𝑛 𝑇 𝑛! 𝑛=0

𝑓 (𝑇) = ∑

(2)

the Hadamard product (or convolution) [1] of 𝑓 and 𝑔 is defined by 1 ∞ + ∑ 𝑎 𝑏 𝑧𝑛 , 𝑧 𝑛=1 𝑛 𝑛

(4)

∞

1 ∞ 𝑔 (𝑧) = + ∑ 𝑏𝑛 𝑧𝑛 , 𝑧 𝑛=1

(𝑓 ∗ 𝑔) (𝑧) =

1 ∫ (𝑧𝐼 − 𝑇)−1 𝑓 (𝑧) 𝑑𝑧, 2𝜋𝑖 𝐶

𝑧 ∈ 𝑈.

(3)

Many subclasses of meromorphic functions have been defined and studied in the past. In particular, the subclasses 𝛼 𝑀𝑙,𝑚1 (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,

(5)

which converges in the norm topology. In this paper, we introduce a subclass 𝑀𝑃 (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 𝑆𝜇𝜃 : Σ → Σ is defined by 𝑆𝜇𝜃 𝑓 (𝑧) =

∞

1 (1 − 𝜇)

𝜃+1

Γ (𝜃 + 1)

∫ 𝑡𝜃+1 𝑒−(𝑡/(1−𝜇)) 𝑓 (𝑧𝑡) 𝑑𝑡, 0

(6)

2

Geometry

then

Conversely, let 𝑆𝜇𝜃 𝑓 (𝑧) =

1 ∞ + ∑ 𝐿 (𝑛, 𝜃, 𝜇) 𝑎𝑛 𝑧𝑛 , 𝑧 𝑛=1

(7)

where 𝐿(𝑛, 𝜃, 𝜇) = (1 − 𝜇)𝑛+1 Γ(𝑛 + 𝜃 + 2)/Γ(𝜃 + 1). Denote by Σ𝑃 the class of all functions 𝑓 ∈ Σ with 𝑎𝑛 ≥ 0. Definition 2. For 0 ≤ 𝜆 < 1, 0 ≤ 𝛼 < 1, a function 𝑓 ∈ Σ𝑃 is in the class 𝑀𝑃 (𝛼, 𝜆, 𝑇) if 󵄩󵄩 󵄩 󵄩󵄩𝑇(𝑆𝜃 𝑓 (𝑇))󸀠 − {(𝜆 − 1) 𝑆𝜃 𝑓 (𝑇) + 𝜆𝑇(𝑆𝜃 𝑓 (𝑇))󸀠 }󵄩󵄩󵄩 𝜇 𝜇 𝜇 󵄩󵄩 󵄩󵄩 󵄩󵄩 󸀠 < 󵄩󵄩󵄩𝑇(𝑆𝜇𝜃 𝑓 (𝑇)) 󵄩 󸀠 󵄩 󵄩 + (1 − 2𝛼) {(𝜆 − 1) 𝑆𝜇𝜃 𝑓 (𝑇) + 𝜆𝑇(𝑆𝜇𝜃 𝑓 (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 ∈ Σ𝑃 given by (1) is in the class 𝑀𝑃 (𝛼, 𝜆, 𝑇) if and only if ∞

∑ [𝑛 + 𝛼 − 𝛼𝜆 (𝑛 + 1)] 𝐿 (𝑛, 𝜃, 𝜇) 𝑎𝑛 ≤ 1 − 𝛼.

(9)

𝑛=1

󵄩 󵄩󵄩 󵄩󵄩𝑇(𝑆𝜃 𝑓 (𝑇))󸀠 − {(𝜆 − 1) 𝑆𝜃 𝑓 (𝑇) + 𝜆𝑇(𝑆𝜃 𝑓 (𝑇))󸀠 }󵄩󵄩󵄩 𝜇 𝜇 𝜇 󵄩󵄩 󵄩󵄩 󵄩󵄩 󸀠 < 󵄩󵄩󵄩𝑇(𝑆𝜇𝜃 𝑓 (𝑇)) 󵄩 󸀠 󵄩 󵄩 + (1 − 2𝛼) {(𝜆 − 1) 𝑆𝜇𝜃 𝑓 (𝑇) + 𝜆𝑇(𝑆𝜇𝜃 𝑓 (𝑇)) }󵄩󵄩󵄩 . 󵄩

(11)

This implies that 󵄩󵄩 ∞ 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∑ [(𝑛 + 1) (1 − 𝜆)] 𝐿 (𝑛, 𝜃, 𝜇) 𝑎𝑛 𝑇𝑛+1 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑛=1 󵄩󵄩 󵄩󵄩 ∞ 󵄩󵄩 ≤ 󵄩󵄩󵄩2 (1 − 𝛼) − ∑ [𝑛 + (1 − 2𝛼) (𝜆 − 1) + 𝜆 (1 − 2𝛼) 𝑛] 󵄩󵄩 𝑛=1 󵄩 󵄩󵄩 󵄩󵄩 × 𝐿 (𝑛, 𝜃, 𝜇) 𝑎𝑛 𝑇𝑛+1 󵄩󵄩󵄩 . 󵄩󵄩 󵄩 (12) Choose 𝑇 = 𝑒𝐼, 0 < 𝑒 < 1. Then, ∑∞ 𝑛=1 [(𝑛 + 1)(1 − [𝑛 + (1 − 2𝛼)(𝜆 − 1) + 𝜆(1 − 𝜆)]𝐿(𝑛, 𝜃, 𝜇)𝑎𝑛 𝑒𝑛 /(2𝑒(1 − 𝛼) − ∑∞ 𝑛=1 2𝛼)𝑛]𝐿(𝑛, 𝜃 ⋅ 𝜇)𝑎𝑛 𝑒𝑛 ) ≤ 1. As 𝑒 → 1, we obtain (9). Corollary 4. If 𝑓 of the form (1) is in 𝑀𝑃 (𝛼, 𝜆, 𝑇), then 𝑎𝑛 ≤

1−𝛼 . [𝑛 + 𝛼 − 𝛼𝜆 (𝑛 + 1)] 𝐿 (𝑛, 𝜃, 𝜇)

(13)

The result is sharp for 𝑓(𝑧) = 1/𝑧 + ((1 − 𝛼)/[𝑛 + 𝛼 − 𝛼𝜆(𝑛 + 1)]𝐿(𝑛, 𝜃, 𝜇))𝑧𝑛 .

The result is sharp for the function 𝑓(𝑧) = 1/𝑧 + ((1 − 𝛼)/[𝑛 + 𝛼 − 𝛼𝜆(𝑛 + 1)]𝐿(𝑛, 𝜃, 𝜇))𝑧𝑛 .

𝑛 Proof. Let 𝑓(𝑇) = 𝑇−1 + ∑∞ 𝑛=1 𝑎𝑛 𝑇 . Assume that (9) holds. Then,

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

󵄩󵄩 󵄩 󵄩󵄩𝑇(𝑆𝜃 𝑓 (𝑇))󸀠 − {(𝜆 − 1) 𝑆𝜃 𝑓 (𝑇) + 𝜆𝑇(𝑆𝜃 𝑓 (𝑇))󸀠 }󵄩󵄩󵄩 𝜇 𝜇 𝜇 󵄩󵄩 󵄩󵄩 󸀠 󵄩󵄩 − 󵄩󵄩󵄩𝑇(𝑆𝜇𝜃 𝑓 (𝑇)) 󵄩

󸀠 󵄩 󵄩 + (1 − 2𝛼) {(𝜆 − 1) 𝑆𝜇𝜃 𝑓 (𝑇) + 𝜆𝑇(𝑆𝜇𝜃 𝑓 (𝑇)) }󵄩󵄩󵄩 󵄩 󵄩󵄩 ∞ 󵄩 󵄩 󵄩󵄩 󵄩󵄩 = 󵄩󵄩󵄩 ∑ [(𝑛 + 1) − 𝜆 (𝑛 + 1)] 𝐿 (𝑛, 𝜇, 𝜃) 𝑎𝑛 𝑇𝑛 󵄩󵄩󵄩 󵄩󵄩𝑛=1 󵄩󵄩 󵄩 󵄩 󵄩󵄩 −1 (10) − 󵄩󵄩󵄩2𝑇 (1 − 𝛼) ∞

− ∑ [𝑛 + (1 − 2𝛼) (𝜆 − 1) + 𝜆 (1 − 2𝛼) 𝑛] 𝑛=1

󵄩 × 𝐿 (𝑛, 𝜇, 𝜃) 𝑎𝑛 𝑇𝑛 󵄩󵄩󵄩󵄩 ∞

≤ ∑ [𝑛 + 𝛼 − 𝛼𝜆 (𝑛 + 1)] 𝐿 (𝑛, 𝜃, 𝜇) 𝑎𝑛 − (1 − 𝛼) 𝑛=1

≤ 0, by (9) , and hence 𝑓 ∈ 𝑀𝑃 (𝛼, 𝜆, 𝑇).

∞ Proof. Suppose that 𝑓(𝑧) = ∑∞ 𝑛=0 𝜇𝑛 𝑓𝑛 (𝑧) = 1/𝑧+∑𝑛=1 𝜇𝑛 ((1− 𝑛 𝛼)/[𝑛 + 𝛼 − 𝛼𝜆(𝑛 + 1)]𝐿(𝑛, 𝜃, 𝜇))𝑧 ; then, we have ∞

∑ [𝑛 + 𝛼 − 𝛼𝜆 (𝑛 + 1)] 𝐿 (𝑛, 𝜃, 𝜇)

𝑛=1

×

𝜇𝑛 (1 − 𝛼) [𝑛 + 𝛼 − 𝛼𝜆 (𝑛 + 1)] 𝐿 (𝑛, 𝜃, 𝜇)

(14)

∞

= ∑ 𝜇𝑛 = 1 − 𝜇0 < 1. 𝑛=1

By Theorem 3, 𝑓 ∈ 𝑀𝑃 (𝛼, 𝜆, 𝑇). Conversely, assume that 𝑓 is in the class 𝑀𝑃 (𝛼, 𝜆, 𝑇); then, by Corollary 4, 𝑎𝑛 ≤

1−𝛼 . [𝑛 + 𝛼 − 𝛼𝜆 (𝑛 + 1)] 𝐿 (𝑛, 𝜃, 𝜇)

(15)

Set 𝜇𝑛 = ([𝑛+𝛼−𝛼𝜆(𝑛+1)]𝐿(𝑛, 𝜃, 𝜇)/(1−𝛼))𝑎𝑛 , 𝑛 = 1, 2, . . . and ∞ 𝜇0 = 1 − ∑∞ 𝑛=1 𝜇𝑛 . Then, 𝑓(𝑧) = ∑𝑛=0 𝜇𝑛 𝑓𝑛 (𝑧) ∈ 𝑀𝑃 (𝛼, 𝜆, 𝑇).

Geometry

3

3. Distortion Bounds

4. Radii Results

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

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

Theorem 6. If 𝑓 ∈ 𝑀𝑃 (𝛼, 𝜆, 𝑇), then

Theorem 8. Let 𝑓 ∈ 𝑀𝑃 (𝛼, 𝜆, 𝑇). Then, 𝑓 is meromorphically starlike of order 𝛿, (0 ≤ 𝛿 < 1) in |𝑧| < 𝑟1 , where

1 1−𝛼 󵄩 󵄩󵄩 − ‖𝑇‖ , 󵄩󵄩𝑓 (𝑇)󵄩󵄩󵄩 ≥ ‖𝑇‖ (1 + 𝛼 − 2𝛼𝜆) (1 − 𝜇)2 (𝜃 + 1) (𝜃 + 2) 1 1−𝛼 󵄩 󵄩󵄩 + ‖𝑇‖ . 󵄩󵄩𝑓 (𝑇)󵄩󵄩󵄩 ≤ ‖𝑇‖ (1 + 𝛼 − 2𝛼𝜆) (1 − 𝜇)2 (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.

𝑟1 = inf[ 𝑛≥1

(1 − 𝛿) (𝑛+𝛼 − 𝛼𝜆 (𝑛+1)) 𝐿 (𝑛, 𝜃, 𝜇) ] (1 − 𝛼) (𝑛 + 2 − 𝛿)

1/(𝑛+1)

,

(22)

𝑛 ≥ 1. 𝑛 Proof. Let 𝑓(𝑇) = 𝑇−1 + ∑∞ 𝑛=1 𝑎𝑛 𝑇 . Since 𝑓 ∈ 𝑀𝑃 (𝛼, 𝜆, 𝑇) is meromorphically starlike of order 𝛿,

󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 𝑇𝑓󸀠 (𝑇) 󵄩󵄩 ≤ 1 − 𝛿. 󵄩󵄩− − 1 󵄩󵄩 󵄩󵄩 𝑓 (𝑇) 󵄩 󵄩

Proof. By Theorem 3,

(23)

Substituting for 𝑓, the above inequality becomes

∞

(1 − 𝛼 − 2𝛼𝜆) 𝐿 (1, 𝜃, 𝜇) ∑ 𝑎𝑛

∞

𝑛=2

∑(

(17)

∞

≤ ∑ [𝑛 + 𝛼 − 𝛼𝜆 (𝑛 + 1)] 𝐿 (𝑛, 𝜃, 𝜇) 𝑎𝑛 ≤ 1 − 𝛼.

𝑛=1

𝑛+2−𝛿 ) ‖𝑇‖𝑛+1 𝑎𝑛 ≤ 1. 1−𝛿

(24)

By Theorem 3,

𝑛=1

∞

[𝑛 + 𝛼 − 𝛼𝜆 (𝑛 + 1)] 𝐿 (𝑛, 𝜃, 𝜇) 𝑎𝑛 ≤ 1. 1−𝛼 𝑛=1

Therefore,

∑

∞

∑ 𝑎𝑛 ≤

𝑛=1

=

1−𝛼 (1 + 𝛼 − 2𝛼𝜆) 𝐿 (1, 𝜃, 𝜇)

Thus, (24) will be true if (18)

1−𝛼 2

(1 + 𝛼 − 2𝛼𝜆) (1 − 𝜇) (𝜃 + 1) (𝜃 + 2)

(

.

∞

1 1 󵄩 󵄩 − ∑ 𝑎𝑛 ‖𝑇‖𝑛 ≤ 󵄩󵄩󵄩𝑓 (𝑇)󵄩󵄩󵄩 ≤ + ∑ 𝑎 ‖𝑇‖𝑛 . ‖𝑇‖ 𝑛=1 ‖𝑇‖ 𝑛=1 𝑛

(19)

Since ‖𝑇‖ < 1, the above inequality becomes ∞

[𝑛 + 𝛼 − 𝛼𝜆 (𝑛 + 1)] 𝐿 (𝑛, 𝜃, 𝜇) 𝑛+2−𝛿 ) ‖𝑇‖𝑛+1 ≤ , (26) 1−𝛿 (1 − 𝛼)

that is,

𝑛 Also, if 𝑓(𝑇) = 𝑇−1 + ∑∞ 𝑛=1 𝑎𝑛 𝑇 , then ∞

(25)

1/(𝑛+1)

. (27)

Theorem 9. Let 𝑓 ∈ 𝑀𝑃 (𝛼, 𝜆, 𝑇). Then, 𝑓 is meromorphically convex of order 𝛿, (0 ≤ 𝛿 < 1) in |𝑧| < 𝑟2 , where

∞

1 1 󵄩 󵄩 − ‖𝑇‖ ∑ 𝑎𝑛 ≤ 󵄩󵄩󵄩𝑓 (𝑇)󵄩󵄩󵄩 ≤ + ‖𝑇‖ ∑ 𝑎𝑛 . ‖𝑇‖ ‖𝑇‖ 𝑛=1 𝑛=1

(1 − 𝛿) (𝑛 + 𝛼 − 𝛼𝜆 (𝑛 + 1)) 𝐿 (𝑛, 𝜃, 𝜇) ] ‖𝑇‖ ≤ [ (1 − 𝛼) (𝑛 + 2 − 𝛿)

(20)

Using (18), we get the result. Theorem 7. If 𝑓 ∈ 𝑀𝑃 (𝛼, 𝜆, 𝑇), then 1 1−𝛼 󵄩 󵄩󵄩 󸀠 󵄩󵄩𝑓 (𝑇)󵄩󵄩󵄩 ≥ 󵄩 ‖𝑇‖2 − (1 + 𝛼 − 2𝛼𝜆) (1 − 𝜇)2 (𝜃 + 1) (𝜃 + 2) , 󵄩 1 1−𝛼 󵄩 󵄩󵄩 󸀠 󵄩󵄩𝑓 (𝑇)󵄩󵄩󵄩 ≤ 󵄩 ‖𝑇‖2 + (1 + 𝛼 − 2𝛼𝜆) (1 − 𝜇)2 (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.

𝑟2 = inf [ 𝑛≥ 1

(1 − 𝛿) (𝑛 + 𝛼 − 𝛼𝜆 (𝑛 + 1)) 𝐿 (𝑛, 𝜃, 𝜇) ] (1 − 𝛼) 𝑛 (𝑛 + 2 − 𝛿)

1/(𝑛+1)

. (28)

Theorem 10. The class 𝑀𝑃 (𝛼, 𝜆, 𝑇) is closed under convex combination. 𝑛 −1 + Proof. Let 𝑓(𝑧) = 𝑇−1 + ∑∞ 𝑛=0 𝑎𝑛 𝑇 and 𝑔(𝑧) = 𝑇 𝑛 𝑏 𝑇 ∈ 𝑀 (𝛼, 𝜆, 𝑇). Then, by Theorem 3, ∑∞ 𝑃 𝑛=0 𝑛 ∞

∑ [𝑛 + 𝛼 − 𝛼𝜆 (𝑛 + 1)] 𝐿 (𝑛, 𝜃, 𝜇) 𝑎𝑛 ≤ 1 − 𝛼,

𝑛=1 ∞

∑ [𝑛 + 𝛼 − 𝛼𝜆 (𝑛 + 1)] 𝐿 (𝑛, 𝜃, 𝜇) 𝑏𝑛 ≤ 1 − 𝛼.

𝑛=1

(29)

4

Geometry

Define ℎ(𝑇) = 𝜆𝑓(𝑇) + (1 − 𝜆)𝑔(𝑇). Then, ℎ(𝑇) = 𝑇−1 + 𝑛 ∑∞ 𝑛=1 [𝜆𝑎𝑛 + (1 − 𝜆)𝑏𝑛 ]𝑇 . Now,

that is, √𝑎𝑛 𝑏𝑛 ≤

∞

∑ [𝑛 + 𝛼 − 𝛼𝜆 (𝑛 + 1)] 𝐿 (𝑛, 𝜃, 𝜇) [𝜆𝑎𝑛 + (1 − 𝜆) 𝑏𝑛 ]

(36)

It is enough to find the largest 𝜉 such that

𝑛=1

∞

= 𝜆 ∑ [𝑛 + 𝛼 − 𝛼𝜆 (𝑛 + 1)] 𝐿 (𝑛, 𝜃, 𝜇) 𝑎𝑛 𝑛=1

(30)

∞

1−𝛼 . [𝑛 + 𝛼 − 𝛼𝜆 (𝑛 + 1)] 𝐿 (𝑛, 𝜃, 𝜇)

+ (1 − 𝜆) ∑ [𝑛 + 𝛼 − 𝛼𝜆 (𝑛 + 1)] 𝐿 (𝑛, 𝜃, 𝜇) 𝑏𝑛

1−𝛼 (1 − 𝜉) [𝑛 + 𝛼 − 𝛼𝜆 (𝑛 + 1)] ≤ [𝑛+𝛼 − 𝛼𝜆 (𝑛 +1)] 𝐿 (𝑛 ⋅ 𝜃, 𝜇) (1 − 𝛼) [𝑛 + 𝜉 − 𝜉𝜆 (𝑛 + 1)] (37) which yields

𝑛=1

𝜉 ≤ ((𝑛 + 𝛼 − 𝛼𝜆 (𝑛 + 1))2 𝐿 (𝑛, 𝜃, 𝜇) − (1 − 𝛼)2 𝑛)

≤ 𝜆 (1 − 𝛼) + (1 − 𝜆) (1 − 𝛼) (using (30)) = 1 − 𝛼.

× ((1 − 𝛼)2 (1 − 𝜆 (𝑛 + 1))

Thus, ℎ ∈ 𝑀𝑃 (𝛼, 𝜆, 𝑇).

(38) −1

+ (𝑛 + 𝛼 − 𝛼𝜆 (𝑛 + 1))2 𝐿 (𝑛, 𝜃, 𝜇)) .

5. Hadamard Product

It follows from (38) that

Theorem 11. The Hadamard product 𝑓 ∗ 𝑔 of two functions 𝑓 and 𝑔 in 𝑀𝑃 (𝛼, 𝜆, 𝑇) belongs to the class 𝑀𝑃 (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).

𝜉 ≤ 1 − (1 − 𝛼)2 (𝑛 + 1) (1 − 𝜆) × ((1 − 𝛼)2 (1 − 𝜆 (𝑛 + 1)) −1

+ (𝑛 + 𝛼 − 𝛼𝜆 (𝑛 + 1))2 𝐿 (𝑛, 𝜃, 𝜇)) .

Proof. It suffices to prove that Let

∞

[𝑛 + 𝜉 − 𝜉𝜆 (𝑛 + 1)] 𝐿 (𝑛, 𝜃, 𝜇) 𝑎𝑛 𝑏𝑛 ≤ 1, ∑ 1−𝜉 𝑛=1

(39)

(31)

𝛿 (𝑛) = (1 − 𝛼)2 (𝑛 + 1) (1 − 𝜆) × ((1 − 𝛼)2 (1 − 𝜆 (𝑛 + 1))

where

−1

+ (𝑛 + 𝛼 − 𝛼𝜆 (𝑛 + 1))2 𝐿 (𝑛, 𝜃, 𝜇)) .

𝜉 = 1 − (2(1 − 𝛼)2 (1 − 𝜆)) × ((1 − 𝛼)2 (1 − 2𝜆) + (1 − 𝛼 − 2𝛼𝜆)2 2

(32)

−1

× (1 − 𝜇) (𝜃 + 1) (𝜃 + 2)) .

[𝑛 + 𝛼 − 𝛼𝜆 (𝑛 + 1)] 𝐿 (𝑛, 𝜃, 𝜇) 𝑎𝑛 ≤ 1, 1−𝛼

∑∞ 𝑛=1 [𝑛 + 𝛼 − 𝛼𝜆 (𝑛 + 1)] 𝐿 (𝑛, 𝜃, 𝜇) 𝑏𝑛 ≤ 1. 1−𝛼

(33)

By Cauchy-Schwartz inequality, ∑∞ 𝑛=1 [𝑛 + 𝛼 − 𝛼𝜆 (𝑛 + 1)] 𝐿 (𝑛, 𝜃, 𝜇) √𝑎𝑛 𝑏𝑛 ≤ 1. 1−𝛼

(34)

We have to find the largest 𝜉 such that [𝑛 + 𝜉 − 𝜉𝜆 (𝑛 + 1)] 𝐿 (𝑛, 𝜃, 𝜇) 𝑎𝑛 𝑏𝑛 1−𝜉 [𝑛 + 𝛼 − 𝛼𝜆 (𝑛 + 1)] 𝐿 (𝑛, 𝜃, 𝜇) √𝑎𝑛 𝑏𝑛 , ≤ 1−𝛼

Clearly 𝛿(𝑛) is an increasing function of 𝑛 (𝑛 ≥ 1). Letting 𝑛 = 1, we prove the assertion.

References

Since 𝑓, 𝑔 ∈ 𝑀𝑃 (𝛼, 𝜆, 𝑇) ∑∞ 𝑛=1

(40)

(35)

[1] P. L. Duren, Univalent Functions, Springer, New York, NY, USA, 1983. [2] T. Rosy, G. Murugusundaramoorthy, and K. Muthunagai, “Certain subclasses of meromorphic functions associated with hypergeometric functions,” International Journal of Nonlinear Science, vol. 12, no. 2, pp. 1749–1759, 2011. [3] G. Murugusundaramoorthy and T. Rosy, “Subclasses of analytic functions associated with Fox-Wright’s generalized hypergeometric functions based on Hilbert space operator,” Studia Universitatis Babes¸-Bolyai, vol. 56, no. 3, pp. 61–72, 2011. [4] N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, Wiley-Interscience, New York, NY, USA, 1958. [5] K. Fan, “Analytic functions of a proper contraction,” Mathematische Zeitschrift, vol. 160, no. 3, pp. 275–290, 1978. [6] W. G. Atshan and R. H. Buti, “Fractional calculus of a class of univalent functions with negative coefficients defined by Hadamard product with Rafid-Operator,” European Journal of Pure and Applied Mathematics, vol. 4, no. 2, pp. 162–173, 2011.

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