Some results for strong differential subordination of analytic functions defined by differential operator

(1)

E-mail address: [email protected] Received May 12, 2019

Available online at http://scik.org Eng. Math. Lett. 2019, 2019:12

https://doi.org/10.28919/eml/4118 ISSN: 2049-9337

SOME RESULTS FOR STRONG DIFFERENTIAL SUBORDINATION OF ANALYTIC FUNCTIONS DEFINED BY DIFFERENTIAL OPERATOR

ASRAA ABDUL JALEEL HUSIEN

Technical Institute, Diwaniya, Al-Furat Al-Awsat Technical University, Iraq

Copyright © 2019 the author(s). 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.

Abstract: The purpose of this paper is to introduce some results associated with the strong differential subordinations of analytic functions defined in the open unit disk and closed unit disk of the complex plane.

Keywords: analytic functions; convex functions; strong differential subordinations; differential operator. 2010 AMS Subject Classification: 30C45.

1. I

NTRODUCTION

Denote by ℋ(𝑈 × 𝑈̅) the family of all analytic functions in 𝑈 × 𝑈̅. Let 𝑈 = {𝑧 ∈ ℂ ∶ |𝑧| <

1} and 𝑈̅ = {𝑧 ∈ ℂ ∶ |𝑧| ≤ 1} indicate the open unit disk and the closed unit disk of the

complex plane, respectively. For 𝑛 ∈ ℕ = {1,2, … } and 𝑎 ∈ ℂ, let ℋ∗_{[𝑎, 𝑛, 𝜁] = {𝑓 ∈}

ℋ(𝑈 × 𝑈̅): 𝑓(𝑧, 𝜁) = 𝑎 + 𝑎𝑛(𝜁)𝑧𝑛 + 𝑎𝑛+1(𝜁)𝑧𝑛+1+ ⋯ , 𝑧 ∈ 𝑈, 𝜁 ∈ 𝑈̅}, where 𝑎𝑘(𝜁) are

holomorphic functions in 𝑈̅ for 𝑘 ≥ 𝑛.

Also, let 𝒜_{𝑛 𝜁}∗ _{= {𝑓 ∈ ℋ(𝑈 × 𝑈̅): 𝑓(𝑧, 𝜁) = 𝑧 + 𝑎}

𝑛+1(𝜁)𝑧𝑛+1+ ⋯ , 𝑧 ∈ 𝑈, 𝜁 ∈ 𝑈̅}, where

𝑎𝑘(𝜁) are holomorphic functions in 𝑈̅ for 𝑘 ≥ 𝑛 + 1.

(2)

𝑅𝑒 {𝑧𝑓𝑧

′_{(𝑧, 𝜁)}

𝑓(𝑧, 𝜁) } > 0 , ( 𝑧 ∈ 𝑈, 𝜁 ∈ 𝑈̅).

Denote the class of all starlike functions in 𝑈 × 𝑈̅ by 𝑆_𝜁∗_.

Similar, 𝑓 ∈ ℋ∗_{[𝑎, 𝑛, 𝜁]}_{is said to be convex in}_{𝑈 × 𝑈̅}_if

𝑅𝑒 {𝑧𝑓𝑧2

′′_{(𝑧, 𝜁)}

𝑓𝑧′(𝑧, 𝜁) + 1} > 0 , ( 𝑧 ∈ 𝑈, 𝜁 ∈ 𝑈̅).

Denote the class of all convex functions in 𝑈 × 𝑈̅ by 𝐾_𝜁∗_.

Definition 1.1 (Oros and Gh Oros, [9]) Let 𝑓(𝑧, 𝜁), 𝑔(𝑧, 𝜁) analytic in 𝑈 × 𝑈̅. The function

𝑓(𝑧, 𝜁) is said to be strongly subordinate to 𝑔(𝑧, 𝜁), written 𝑓(𝑧, 𝜁) ≺≺ 𝐹(𝑧, 𝜁), 𝑧 ∈ 𝑈, 𝜁 ∈ 𝑈̅,

if there exists an analytic function 𝑤 in 𝑈 with 𝑤(0) = 0 and |𝑤(𝑧)| < 1, 𝑧 ∈ 𝑈 such that

𝑓(𝑧, 𝜁) = 𝑔(𝑤(𝑧), 𝜁) for all 𝜁 ∈ 𝑈̅.

Remark 1.1 (Oros and Gh Oros, [9])

1) Since 𝑓(𝑧, 𝜁) is analytic in 𝑈 × 𝑈̅, for all 𝜁 ∈ 𝑈̅ and univalent in 𝑈, for all 𝜁 ∈ 𝑈̅,

Definition1.1 is equivalent to 𝑓(0, 𝜁) = 𝑔(0, 𝜁) for all 𝜁 ∈ 𝑈̅ and 𝑓(𝑈 × 𝑈̅) ⊂ 𝑔(𝑈 × 𝑈̅).

2) If 𝑓(𝑧, 𝜁) = 𝑓(𝑧) and 𝑔(𝑧, 𝜁) = 𝑔(𝑧), the strong subordination becomes the usual notion

of subordination.

Let 𝒜_𝜁∗ denote the subclass of the functions 𝑓(𝑧, 𝜁) ∈ ℋ(𝑈 × 𝑈̅) of the form:

𝑓(𝑧, 𝜁) = 𝑧 + ∑ 𝑎_𝑘(𝜁)𝑧𝑘 ∞

𝑘=2

, 𝑧 ∈ 𝑈, 𝜁 ∈ 𝑈̅ (1.1)

which are analytic and univalent in 𝑈 × 𝑈̅.

For 𝜂 ∈ ℕ₀ = ℕ ∪ {0} 𝛼, 𝜏 ≥ 0, 𝜇, 𝜆, 𝛽 > 0 and 𝛼 ≠ 𝜆, we consider the differential operator

𝐴_{𝜇,𝜆,𝜏}𝜂 (𝛼, 𝛽) ∶ 𝒜_𝜁∗ _{⟶ 𝒜}

𝜁∗, introduced by Amourah and Darus [2], where

𝐴_{𝜇,𝜆,𝜏}𝜂 (𝛼, 𝛽)𝑓(𝑧) = 𝑧 + ∑ [1 +(𝑛 − 1)[(𝜆 − 𝛼)𝛽 + 𝑛𝜏]

𝜇 + 𝜆 ]

𝜂

𝑎_𝑘(𝜁)𝑧𝑘 ∞

𝑘=2

. (1.2)

It is readily verified from (1.2) that

𝑧 (𝐴_{𝜇,𝜆,𝜏}𝜂 (𝛼, 𝛽)𝑓(𝑧, 𝜁))

(3)

= 𝜇 + 𝜆

(𝜆 − 𝛼)𝛽 + 𝑛𝜏𝐴𝜇,𝜆,𝜏

𝜂+1_{(𝛼, 𝛽)𝑓(𝑧, 𝜁) − (1 −} 𝜇 + 𝜆

(𝜆 − 𝛼)𝛽 + 𝑛𝜏) 𝐴𝜇,𝜆,𝜏

𝜂 _{(𝛼, 𝛽)𝑓(𝑧, 𝜁). (1.3)}

Some of the special cases of the operator defined by (1.2) can be found in [1,4,10,11].

In recent years, many authors obtained various interesting results associated with strong

differential subordination and superordination for example (see [3,5,6,12,13,14]).

In order to derive our main results, we need the following Lemmas.

Lemma 1.1 (Miller and Mocanu, [8]) Let ℎ(𝑧, 𝜁) be a convex function with ℎ(0, 𝜁) = 𝑎, for

every 𝜁 ∈ 𝑈̅ and let 𝛾 ∈ ℂ∗ _{= ℂ ∖ {0}}_with_{𝑅𝑒 (𝛾) ≥ 0}_{. If}_{𝑝 ∈ ℋ}∗_{[𝑎, 𝑛, 𝜁]}_and

𝑝(𝑧, 𝜁) +1

𝛾𝑧𝑝𝑧′(𝑧, 𝜁) ≺≺ ℎ(𝑧, 𝜁), (𝑧 ∈ 𝑈, 𝜁 ∈ 𝑈̅), (1.4)

then

𝑝(𝑧, 𝜁) ≺≺ 𝑞(𝑧, 𝜁) ≺≺ ℎ(𝑧, 𝜁), (𝑧 ∈ 𝑈, 𝜁 ∈ 𝑈̅),

where 𝑞(𝑧, 𝜁) = 𝛾

𝑛𝑧𝛾𝑛∫ 𝑡 𝛾

𝑛−1ℎ(𝑡, 𝜁)𝑑𝑡

𝑧

0 is convex and it is the best dominant of (1.4).

Lemma 1.2 (Miller and Mocanu, [7]) Let 𝑞(𝑧, 𝜁) be a convex function in 𝑈 × 𝑈̅ for all 𝜁 ∈

𝑈̅ and let ℎ(𝑧, 𝜁) = 𝑞(𝑧, 𝜁) + 𝑛𝛿𝑧𝑞𝑧′(𝑧, 𝜁), 𝑧 ∈ 𝑈, 𝜁 ∈ 𝑈̅, where 𝛿 > 0 and 𝑛 is a positive

integer. If

𝑝(𝑧, 𝜁) = 𝑞(0, 𝜁) + 𝑝𝑛(𝜁)𝑧𝑛 + 𝑝𝑛+1(𝜁)𝑧𝑛+1+ ⋯,

is analytic in 𝑈 × 𝑈̅ and

𝑝(𝑧, 𝜁) + 𝛿𝑧𝑝_𝑧′_{(𝑧, 𝜁) ≺≺ ℎ(𝑧, 𝜁), (𝑧 ∈ 𝑈, 𝜁 ∈ 𝑈̅),}

then

𝑝(𝑧, 𝜁) ≺≺ 𝑞(𝑧, 𝜁), (𝑧 ∈ 𝑈, 𝜁 ∈ 𝑈̅),

and this result is sharp.

2. M

AIN RESULT

Theorem 2.1. Let ℎ(𝑧, 𝜁) be a convex function such that ℎ(0, 𝜁) = 1. If 𝑓 ∈ 𝒜_𝜁∗_{satisfies the}

strong differential subordination:

(𝐴_{𝜇,𝜆,𝜏}𝜂 (𝛼, 𝛽)𝑓(𝑧, 𝜁))

𝑧 ′

(4)

then

𝐴_{𝜇,𝜆,𝜏}𝜂 (𝛼, 𝛽)𝑓(𝑧, 𝜁)

𝑧 ≺≺ 𝑞(𝑧, 𝜁) ≺≺ ℎ(𝑧, 𝜁),

where 𝑞(𝑧, 𝜁) =1_𝑧∫ ℎ(𝑡, 𝜁)𝑑𝑡₀𝑧 is convex and it is the best dominant

Proof. Suppose that

𝑝(𝑧, 𝜁) = 𝐴𝜇,𝜆,𝜏

𝜂 _{(𝛼, 𝛽)𝑓(𝑧, 𝜁)}

𝑧 , 𝑧 ∈ 𝑈, 𝜁 ∈ 𝑈̅ . (2.2)

Then the function 𝑝(𝑧, 𝜁) is analytic in 𝑈 × 𝑈̅ and 𝑝(0, 𝜁) = 1.

Simple computations from (2.2), we get

𝑝(𝑧, 𝜁) + 𝑧𝑝_𝑧′_{(𝑧, 𝜁) = (𝐴}

𝜇,𝜆,𝜏𝜂 (𝛼, 𝛽)𝑓(𝑧, 𝜁))_𝑧 ′

. (2.3)

Using (2.3), (2.1) becomes

𝑝(𝑧, 𝜁) + 𝑧𝑝𝑧′(𝑧, 𝜁) ≺≺ ℎ(𝑧, 𝜁).

An application of Lemma 1.1 with 𝑛 = 1, 𝛾 = 1 yields

𝑧 ≺≺ 𝑞(𝑧, 𝜁) = 1

𝑧∫ ℎ(𝑡, 𝜁)𝑑𝑡

𝑧

0

≺≺ ℎ(𝑧, 𝜁).

By taking ℎ(𝑧, 𝜁) =𝜁+(2𝜌−𝜁)𝑧_1+𝑧 , 0 ≤ 𝜌 < 1 in Theorem 2.1, we obtain the following corollary:

Corollary 2.1. If 𝑓 ∈ 𝒜_𝜁∗_{satisfies the strong differential subordination:}

(𝐴_{𝜇,𝜆,𝜏}𝜂 (𝛼, 𝛽)𝑓(𝑧, 𝜁))

𝑧 ′

≺≺𝜁 + (2𝜌 − 𝜁)𝑧 1 + 𝑧 ,

then

𝑧 ≺≺

1 𝑧∫

𝜁 + (2𝜌 − 𝜁)𝑡 1 + 𝑡 𝑑𝑡

𝑧

0

= 2𝜌 − 𝜁 +2(𝜁 − 𝜌)

𝑧 ln(1 + 𝑧).

Theorem 2.2. Let 𝑞(𝑧, 𝜁) be a convex function such that 𝑞(0, 𝜁) = 1 and let ℎ be the

function ℎ(𝑧, 𝜁) = 𝑞(𝑧, 𝜁) + 𝑧𝑞_𝑧′_{(𝑧, 𝜁)}_{. If}_{𝑓 ∈ 𝒜}

𝜁

∗_{satisfies the strong differential subordination:}

(𝑧 𝐴𝜇,𝜆,𝜏

𝜂+1_{(𝛼, 𝛽)𝑓(𝑧, 𝜁)}

𝐴_{𝜇,𝜆,𝜏}𝜂 (𝛼, 𝛽)𝑓(𝑧, 𝜁) )

𝑧 ′

≺≺ ℎ(𝑧, 𝜁), (2.4)

(5)

𝐴_{𝜇,𝜆,𝜏}𝜂+1(𝛼, 𝛽)𝑓(𝑧, 𝜁)

𝐴_{𝜇,𝜆,𝜏}𝜂 (𝛼, 𝛽)𝑓(𝑧, 𝜁)≺≺ 𝑞(𝑧, 𝜁).

𝑝(𝑧, 𝜁) =𝐴𝜇,𝜆,𝜏

𝜂+1_{(𝛼, 𝛽)𝑓(𝑧, 𝜁)}

𝐴_{𝜇,𝜆,𝜏}𝜂 (𝛼, 𝛽)𝑓(𝑧, 𝜁) , 𝑧 ∈ 𝑈, 𝜁 ∈ 𝑈̅ . (2.5)

Differentiating both sides of (2.5) with respect to 𝑧 and using (2.4), we have

𝑝(𝑧, 𝜁) + 𝑧𝑝𝑧′(𝑧, 𝜁)

=𝐴𝜇,𝜆,𝜏

𝜂+1_{(𝛼, 𝛽)𝑓(𝑧, 𝜁)}

+𝐴𝜇,𝜆,𝜏

𝜂 _{(𝛼, 𝛽)𝑓(𝑧, 𝜁) (𝐴}

𝜇,𝜆,𝜏𝜂+1(𝛼, 𝛽)𝑓(𝑧, 𝜁))_𝑧 ′

− 𝐴_{𝜇,𝜆,𝜏}𝜂+1(𝛼, 𝛽)𝑓(𝑧, 𝜁) (𝐴_{𝜇,𝜆,𝜏}𝜂 (𝛼, 𝛽)𝑓(𝑧, 𝜁))

𝑧 ′

[𝐴_{𝜇,𝜆,𝜏}𝜂 (𝛼, 𝛽)𝑓(𝑧, 𝜁)]2

=𝐴𝜇,𝜆,𝜏

𝜂 _{(𝛼, 𝛽)𝑓(𝑧, 𝜁) (𝑧 𝐴} 𝜇,𝜆,𝜏

𝜂+1_{(𝛼, 𝛽)𝑓(𝑧, 𝜁))} 𝑧 ′

− 𝑧𝐴_{𝜇,𝜆,𝜏}𝜂+1(𝛼, 𝛽)𝑓(𝑧, 𝜁) (𝐴_{𝜇,𝜆,𝜏}𝜂 (𝛼, 𝛽)𝑓(𝑧, 𝜁))

𝑧 ′

[𝐴_{𝜇,𝜆,𝜏}𝜂 (𝛼, 𝛽)𝑓(𝑧, 𝜁)]2

= (𝑧 𝐴𝜇,𝜆,𝜏

𝜂+1_{(𝛼, 𝛽)𝑓(𝑧, 𝜁)}

𝐴_{𝜇,𝜆,𝜏}𝜂 (𝛼, 𝛽)𝑓(𝑧, 𝜁) )

𝑧 ′

≺≺ ℎ(𝑧, 𝜁).

An application of Lemma 1.2, we obtain

𝐴_{𝜇,𝜆,𝜏}𝜂+1(𝛼, 𝛽)𝑓(𝑧, 𝜁)

𝐴_{𝜇,𝜆,𝜏}𝜂 (𝛼, 𝛽)𝑓(𝑧, 𝜁)≺≺ 𝑞(𝑧, 𝜁).

Theorem 2.3. Let ℎ(𝑧, 𝜁) be a convex function such that ℎ(0, 𝜁) = 1. If 0 ≤ 𝜎 < 𝑝, 𝜃 ∈ ℂ

and 𝑓 ∈ 𝒜_𝜁∗_{satisfies the strong differential subordination:}

1 − 𝜃 1 − 𝜎 (

𝑧 − 𝜎) +

𝜃

1 − 𝜎 ((𝐴𝜇,𝜆,𝜏𝜂 (𝛼, 𝛽)𝑓(𝑧, 𝜁))_𝑧 ′

− 𝜎) ≺≺ ℎ(𝑧, 𝜁), (2.6)

then

1 1 − 𝜎 (

(6)

where 𝑞(𝑧, 𝜁) =1_𝜃𝑧− 1𝜃∫ 𝑡 1

𝜃−1ℎ(𝑡, 𝜁)𝑑𝑡

𝑧

0 is convex and it is the best dominant.

𝑝(𝑧, 𝜁) = 1 1 − 𝜎 (

𝑧 − 𝜎) , 𝑧 ∈ 𝑈, 𝜁 ∈ 𝑈̅ . (2.7)

Differentiating both sides of (2.7) with respect to 𝑧, we have

𝑝(𝑧, 𝜁) + 𝜃𝑧𝑝_𝑧′_{(𝑧, 𝜁)}

=1 − 𝜃 1 − 𝜎 (

𝑧 − 𝜎) +

𝜃

1 − 𝜎 ((𝐴𝜇,𝜆,𝜏𝜂 (𝛼, 𝛽)𝑓(𝑧, 𝜁))_𝑧 ′

− 𝜎). (2.8)

From (2.6) and (2.8), we get

𝑝(𝑧, 𝜁) + 𝜃𝑧𝑝𝑧′(𝑧, 𝜁) ≺≺ ℎ(𝑧, 𝜁).

An application of Lemma 1.1 with 𝑛 = 1, 𝛾 =_𝜃1 yields

1 1 − 𝜎 (

𝑧 − 𝜎) ≺≺ 𝑞(𝑧, 𝜁) = 1 𝜃𝑧

− _𝜃1_{∫ 𝑡}𝑧 1_𝜃−1_{ℎ(𝑡, 𝜁)𝑑𝑡}

0

≺≺ ℎ(𝑧, 𝜁).

Theorem 2.4. Let 𝑞(𝑧, 𝜁) be a convex function such that 𝑞(0, 𝜁) = 1 and let ℎ be the

function ℎ(𝑧, 𝜁) = 𝑞(𝑧, 𝜁) + 1

1+_{(𝜆−𝛼)𝛽+𝑛𝜏}𝜇+𝜆 𝑧𝑞𝑧

′_{(𝑧, 𝜁)}_{, where} 𝜇+𝜆

(𝜆−𝛼)𝛽+𝑛𝜏> 0. Suppose that

𝐹(𝑧, 𝜁) =

1 +_{(𝜆 − 𝛼)𝛽 + 𝑛𝜏}𝜇 + 𝜆

𝑧(𝜆−𝛼)𝛽+𝑛𝜏𝜇+𝜆

∫ 𝑡𝑧 (𝜆−𝛼)𝛽+𝑛𝜏−1𝜇+𝜆 _{𝑓(𝑡, 𝜁)𝑑𝑡} 0

, 𝑧 ∈ 𝑈, 𝜁 ∈ 𝑈̅. (2.9)

If 𝑓 ∈ 𝒜_𝜁∗_(𝑝)_{satisfies the strong differential subordination}

(𝐴_{𝜇,𝜆,𝜏}𝜂 (𝛼, 𝛽)𝑓(𝑧, 𝜁))

𝑧 ′

≺≺ ℎ(𝑧, 𝜁), (2.10)

then

(𝐴_{𝜇,𝜆,𝜏}𝜂 (𝛼, 𝛽)𝐹(𝑧, 𝜁))

𝑧 ′

≺≺ 𝑞(𝑧, 𝜁).

𝑝(𝑧, 𝜁) = (𝐴_{𝜇,𝜆,𝜏}𝜂 (𝛼, 𝛽)𝐹(𝑧, 𝜁))

𝑧 ′

(7)

From (2.9), we have

𝑧(𝜆−𝛼)𝛽+𝑛𝜏𝜇+𝜆 _{𝐹(𝑧, 𝜁) = (1 +} 𝜇 + 𝜆

(𝜆 − 𝛼)𝛽 + 𝑛𝜏) ∫ 𝑡

𝜇+𝜆

(𝜆−𝛼)𝛽+𝑛𝜏−1_{𝑓(𝑡, 𝜁)𝑑𝑡} 𝑧

0

. (2.12)

Differentiating both sides of (2.12) with respect to 𝑧, we get

(1 + 𝜇 + 𝜆

(𝜆 − 𝛼)𝛽 + 𝑛𝜏) 𝑓(𝑧, 𝜁) =

𝜇 + 𝜆

(𝜆 − 𝛼)𝛽 + 𝑛𝜏𝐹(𝑧, 𝜁) + 𝑧𝐹𝑧′(𝑧, 𝜁)

and

(1 + 𝜇 + 𝜆

(𝜆 − 𝛼)𝛽 + 𝑛𝜏) 𝐴𝜇,𝜆,𝜏𝜂 (𝛼, 𝛽)𝑓(𝑧, 𝜁)

= 𝜇 + 𝜆

(𝜆 − 𝛼)𝛽 + 𝑛𝜏𝐴𝜇,𝜆,𝜏𝜂 (𝛼, 𝛽)𝐹(𝑧, 𝜁) + 𝑧 (𝐴𝜇,𝜆,𝜏𝜂 (𝛼, 𝛽)𝐹(𝑧, 𝜁))_𝑧 ′

.

So

(𝐴_{𝜇,𝜆,𝜏}𝜂 (𝛼, 𝛽)𝑓(𝑧, 𝜁))

𝑧 ′

= (𝐴_{𝜇,𝜆,𝜏}𝜂 (𝛼, 𝛽)𝐹(𝑧, 𝜁))

𝑧 ′

+𝑧 (𝐴𝜇,𝜆,𝜏

𝜂 _{(𝛼, 𝛽)𝐹(𝑧, 𝜁))} 𝑧2

′′

1 +_{(𝜆 − 𝛼)𝛽 + 𝑛𝜏}𝜇 + 𝜆 . (2.13)

From (2.11) and (2.13), we obtain

𝑝(𝑧, 𝜁) + 1

1 +_{(𝜆 − 𝛼)𝛽 + 𝑛𝜏}𝜇 + 𝜆 𝑧𝑝𝑧

′_{(𝑧, 𝜁) = (𝐴} 𝜇,𝜆,𝜏

𝜂 _{(𝛼, 𝛽)𝑓(𝑧, 𝜁))} 𝑧 ′

. (2.14)

Using (2.14), (2.10) becomes

𝑝(𝑧, 𝜁) + 1

1 +_{(𝜆 − 𝛼)𝛽 + 𝑛𝜏}𝜇 + 𝜆 𝑧𝑝𝑧

′_{(𝑧, 𝜁) ≺≺ 𝑞(𝑧, 𝜁) +} 1

1 +_{(𝜆 − 𝛼)𝛽 + 𝑛𝜏}𝜇 + 𝜆 𝑧𝑞𝑧

′_{(𝑧, 𝜁).}

An application of Lemma 1.2 yields 𝑝(𝑧, 𝜁) ≺≺ 𝑞(𝑧, 𝜁). By using (2.10), we obtain

(𝐴_{𝜇,𝜆,𝜏}𝜂 (𝛼, 𝛽)𝐹(𝑧, 𝜁))

𝑧 ′

≺≺ 𝑞(𝑧, 𝜁).

Conflict of Interests

(8)

REFERENCES

[1] F. M. Al-Oboudi, On univalent functions defined by a generalized Salagean operator, Int. J. Math. Math. Sci., 27(2004), 1429–1436.

[2] A. Amourah and M. Darus, Some properties of a new class of univalent functions involving a new generalized differential operator with negative coefficients, Indian J. Sci. Tech., 9(36)(2016), 1–7.

[3] N. E. Cho, O. S. Kwon and H. M. Srivastava, Strong differential subordination and superordination for multivalently meromorphic functions involving the Liu- Srivastava operator, Integral transforms Spec. Funct., 21(8)(2010), 589-601.

[4] M. Darus and R. W. Ibrahim, On subclasses for generalized operators of complex order, Far East J. Math. Sci., 33(3)(2009), 299–308.

[5] M. P. Jeyaraman and T. K. Suresh, Strong differential subordination and superordination of analytic functions, J. Math. Anal. Appl., 385(2)(2012), 854-864.

[6] A. A. Lupas, A note on strong differential superordinations using a generalized Salagean operator and Ruscheweyh operator, Stud. Univ. Babes-Bolyai Math., 57(2) (2012), 135-165.

[7] S. S. Miller and P. T. Mocanu, On some classes of first-order differential subordinations, Michigan Math. J., 32(2)(1985), 185-195.

[8] S. S. Miller and P. T. Mocanu, Differential Subordinations: Theory and Applications, Series on Monographs and Textbooks in Pure and Applied Mathematics Vol. 225, Marcel Dekker Inc., New York and Basel, 2000. [9] G. I. Oros and Gh. Oros, Strong differential subordination, Turk. J. Math., 33(2009), 249-257.

[10] G. S. Salagean, Subclasses of univalent functions, Lecture Notes in Math., Springer Verlag, Berlin, 1013(1983), 362-372.

[11] A. K. Wanas and A. Alb Lupas, On a new strong differential subordinations and superordinations of analytic functions involving the generalized differential operator, Int. J. Pure Appl. Math., 116(3)(2017), 571-579. [12] A. K. Wanas and A. H. Majeed, New strong differential subordinations and superordinations of symmetric

analytic functions, Int. J. Math. Anal., 11(11)(2017), 543-549.

(9)