A new subclass of meromorphic functions defined by Rapid operator

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C om mun.Fac.Sci.U niv.A nk.Ser. A 1 M ath. Stat.

Volum e 70, N umb er 2, Pages 858–870 (2021) D O I: 10.31801/cfsuasm as.835272

ISSN 1303–5991 E-ISSN 2618–6470

Received by the editors: D ecem ber 3, 2020; Accepted: M arch 7, 2021

A NEW SUBCLASS OF MEROMORPHIC FUNCTIONS DEFINED BY RAPID OPERATOR

Venkateswarlu BOLINENI¹, Thirupathi Reddy PINNINTI², SUJATHA¹, and Sridevi SETTIPALLI¹

1Department of Mathematics, GSS, GITAM University, Doddaballapur- 562 163, Bengaluru Rural, Karnataka, INDIA

2Department of Mathematics, Kakatiya University,Warangal- 506 009, Telangana, INDIA

Abstract. We present and investigate a new subclass of meromorphic univalent functions described by the Rapid operator in this study. Coe¢ cient inequalities is discussed, as well as distortion properties, closure theorems, Hadamard product. After this, integral transforms for the class (#; %; }; ; ) are obtained.

1. Introduction Let stands for the function class of the form

@(~) = 1

~+ X1

`=1

a`~^`; ` 2 N = f1; 2; 3; g (1) analytic in the punctured unit disc = f~ 2 C : 0 < j~j < 1g = n f0g:

A function @ 2 given by (1) is said to be meromorphically starlike of order % if it satis…es the following:

< ~@⁰(~)

@(~) > %; (~ 2 )

for some %(0 % < 1): We say that @ is in the class (%) of such functions.

Similarly a function @ 2 given by (1) is said to be meromorphically convex of order % if it satis…es the following:

< 1 + ~@⁰⁰(~)

@⁰(~) > %; (~ 2 )

2020 Mathematics Subject Classi…cation. 30C45.

Keywords and phrases. Meromorphic, starlike, coe¢ cient estimates, integral operator.

bvlmaths@gmail.com-Corresponding author; reddypt2@gmail.com, sujathavaish- navy@gmail.com; siri_settipalli@yahoo.co.in

0000-0003-3669-350X; 0000-0002-0034-444X; 0000-0002-2109-3328; 0000-0003-1918-6127.

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858

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for some %(0 % < 1): We say that @ is in the class `(%) of such functions.

Akgul [1,2], Miller [8], Pommerenke [9], Royster [10], Aydogan and Sakar [4,5,11]

and Venkateswarlu et al. [14, 15, 16] have all studied the class (%) and numerous other subclasses of extensively.

For functions @ 2 given by (1) and g 2 given by g(~) = 1

~ + X1

`=1

b_`~^`; we de…ne the Hadamard product of @ and g by

(@ g)(~) = 1

~+ X1

`=1

a_`b_`~^`:

Jung et al. de…ned the integral operator on normalised analytic functions in [6]

and Lashin [7] updated their operator for meromorphic functions in the following manner:

Lemma 1. For @ 2 given by (1), if the operator S : ! is de…ned by

S @(~) = 1

(1 ) ( + 1) Z1

0

t ⁺¹e¹ ^t @(t~)dt; (2)

(0 < 1; 0 1 and ~ 2 ) then S @(~) = 1

~ + X1

`=1

`( ; )a`~^` (3)

where _`( ; ) = (1 )^`+1 ^{(`+ +2)}_{( +1)} and is the familiar Gamma function.

Using the equation (3), it is easily seen that

~(S @(~))⁰= S ¹@(~) ( + 1)S @(~); (0 1; 0 1): (4) We de…ne a new subclass (#; %; }; ; ) of based on Sivaprasad Kumar et al. [13] and Venkateswarlu et al. [14] (#; %; }; ; ) of :

De…nition 2. For 0 # < 1; % 0; 0 } < ¹₂; we let (#; %; }; ; ) be the subclass of consisting of functions of the form (1) and satisfying the analytic condition

< ~(S @(~))⁰²(S @(~))⁰⁰

(1 })S @(~) + }~(S @(~))⁰ + #

!

> % ~(S @(~))⁰²(S @(~))⁰⁰

(1 })S @(~) + }~(S @(~))⁰ + 1 : (5) The following lemmas are needed to prove our …ndings [3].

Lemma 3. If is a real number and ! is a complex number then

<(!) , j! + (1 )j j! (1 + )j 0:

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Lemma 4. If ! is a complex number and ; ` are real numbers then

<(!) `j! + 1j + , < !(1 + `eⁱ ) + `eⁱ ; ( ):

The key purpose of this paper is to look at some traditional geometric function theory properties for the class of geometric functions, such as coe¢ cient bounds, distortion properties, closure theorems, Hadamard product, and integral transforms.

2. Coefficient estimates

We obtain required and adequate conditions for a function @ to be in the class in this section.

Theorem 5. Let @ 2 be given by (1). Then @ 2 (#; %; }; ; ) i¤

X1

`=1

[(1 + (` 1)})][`(% + 1) + (% + #)] _`( ; )a` (1 #)(1 2}): (6) Proof. Let @ 2 (#; %; }; ; ): Then by De…nition 2 and using Lemma 4, It su¢ ces to demonstrate that

<

( ~(S @(~))⁰²(S @(~))⁰⁰

(1 })S @(~) + }~(S @(~))⁰(1 + %eⁱ ) + %eⁱ )

#; ( ): (7)

For convenience

C(~) = ~(S @(~))⁰²(S @(~))⁰⁰ (1 + %eⁱ )

%eⁱ (1 })S @(~) + }~(S @(~))⁰ D(~) =(1 })S @(~) + }~(S @(~))⁰:

That is, the equation (7) is equivalent to

< C(~)

D(~) #:

We only need to prove that in light of Lemma 3

jC(~) + (1 #)D(~)j jC(~) (1 + #)D(~)j 0:

Therefore

jC(~) + (1 #)D(~)j (2 #)(1 2}) 1

j~j X1

`=1

[` (1 #)][1 + }(` 1)] _`( ; )a`j~j^`

% X1

`=1

(` + 1)[1 + }(` 1)] _`( ; )a_`j~j^` and jC(~) (1 + #)D(~)j

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#(1 2}) 1 j~j +

X1

`=1

[` + (1 + #)][1 + }(` 1)] _`( ; )a_`j~j^`

+ % X1

`=1

(` + 1)[1 + }(` 1)] _`( ; )a_`j~j^`:

It is to show that

jC(~) + (1 #)D(~)j jC(~) (1 + #)D(~)j 2(1 #)(1 2}) 1

j~j 2 X1

`=1

[(` + #)(1 + (` 1)})] _`( ; )a`j~j^`

2%

X1

`=1

(` + 1)(1 + (` 1)}) _`( ; )a`j~j^` 0; by the given condition (6).

Conversely suppose @ 2 (#; %; }; ; ): Then by Lemma 3, we have (7).

The inequality (7) is reduced to when the values of ~ are chosen on the positive real axis

<

8>

><

>>

:

[(1 2})(1 #)(1 + %eⁱ )]¹

~² +P¹

`=1f` + %eⁱ (` + 1) + #g[1 + }(` 1)] _`( ; )~^{` 1} (1 2})_~¹2 +P¹

`=1

[1 + }(` 1)] _`( ; )a`~^{` 1}

9>

>=

>>

; 0:

Since <( eⁱ ) jeⁱ j = 1; the above inequality is reduced to

<

8>

><

>>

:

[(1 2})(1 #)(1 + %eⁱ )]_r¹2 +P¹

`=1f` + %(` + 1) + #g[1 + }(` 1)] _`( ; )a`r^{` 1} (1 2})_r¹2 +P¹

`=1

[1 + }(` 1)] _`( ; )r^{` 1}

9>

>=

>>

; 0:

We obtained the inequality (6) by letting r ! 1 and using the mean value theorem.

Corollary 6. If @ 2 (#; %; }; ; ) then

a_` (1 #)(1 2})

[1 + }(` 1)][`(1 + %) + (# + %)] _`( ; ): (8) The estimate is sharp for the function

@(~) = 1

~+ (1 #)(1 2})

[1 + }(` 1)][`(1 + %) + (# + %)] _`( ; )~^`: (9) We get the following corollary by taking } = 0 in Theorem 5.

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Corollary 7. If @ 2 (#; %; ; ) then a`

1 #

[`(1 + %) + (# + %)] _`( ; ): (10) 3. Distortion theorem

Theorem 8. If @ 2 (#; %; }; ; ) then for 0 < j~j = r < 1;

1 r

(1 #)(1 2})

(2% + # + 1) ₁( ; ) r j@(~)j 1

r + (1 #)(1 2})

(2% + # + 1) ₁( ; )r: (11) This estimate is sharp for the function

@(~) = 1

~+ (1 #)(1 2})

(2% + # + 1) ₁( ; ) ~: (12)

Proof. Since @(~) = _~¹+P¹

`=1

a`~^`; we have

j@(~)j = 1 r +

X1

`=1

a`r^` 1 r+ r

X1

`=1

a`: (13)

Since ` 1; (2% + # + 1) ₁( ; ) [1 + }(` 1)][`(1 + %) + (% + #)] _`( ; ); using Theorem 5, we have

(2% + # + 1) ₁( ; ) X1

`=1

a`

X1

`=1

[1 + }(` 1)][`(1 + %) + (% + #)] _`( ; ) (1 #)(1 2})

) X1

`=1

a`

(1 #)(1 2}) (2% + # + 1) ₁( ; ): Using the above inequality in (13), we have

j@(~)j 1

r + (1 #)(1 2}) (2% + # + 1) ₁( ; )r and j@(~)j 1

r

(1 #)(1 2}) (2% + # + 1) ₁( ; )r:

The estimate is sharp for the function @(~) = ¹_~+ (1 #)(1 2}) (2%+#+1) ₁( ; )~:

We omit the proof of the following corollary since it is similar to that of Theorem 8.

Corollary 9. If @ 2 (#; %; }; ; ) then 1

r²

(1 #)(1 2})

(2% + # + 1) ₁( ; ) j@⁰(~)j 1

r² + ((1 #)(1 2}) (2% + # + 1) ₁( ; ): The estimate is sharp for the function given by (12).

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4. Closure theorems Let the function @j be de…ned, for j = 1; 2; ; m; by

@^j(~) = 1

~+ X1

`=1

a`;j~^`; a`;j 0: (14)

Theorem 10. Let the functions @j; j = 1; 2; ; m de…ned by (14) be in the class (#; %; }; ; ): Then the function h de…ned by

h(~) = 1

~+ X1

`=1

0

@1 m

Xm j=1

a`;j

1

A ~^` (15)

also belongs to the class (#; %; }; ; ):

Proof. Since @j; j = 1; 2; ; m are in the class (#; %; }; ; ); it follows from Theorem 5, that

X1

`=1

[1 + }(` 1)][`(1 + %) + (% + #)] _`( ; )a`;j (1 #)(1 2});

for every j = 1; 2; ; m: Hence X1

`=1

[1 + }(` 1)][`(1 + %) + (% + #)] _`( ; ) 0

@1 m

Xm j=1

a_`;j 1 A

= 1 m

Xm j=1

X1

`=1

[1 + }(` 1)][`(1 + %) + (% + #)] _`( ; )a_`;j

!

(1 #)(1 2}):

From Theorem (6), it follows that h 2 (#; %; }; ; ):

Hence the proof.

Theorem 11. The class (#; %; }; ; ) is closed under convex linear combina- tions.

Proof. Let the functions @j; j = 1; 2; ; m de…ned by (14) be in the class (#; %; }; ; ):

Then one need only show that function

h(~) = &@¹(~) + (1 &)@²(~); 0 & 1 (16) is in the class (#; %; }; ; ): Since for 0 & 1;

h(~) = 1

~ + X1

`=1

[&a_`;1+ (1 &)a_`;1]~^`; (17)

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with the assistance of the Theorem5, we have X1

`=1

[1 + }(` 1)][`(1 + %) + (% + #)] _`( ; )[&a`;1+ (1 &)a`;1]

&(1 #)(1 2}) + (1 &)(1 #)(1 2})

=(1 #)(1 2});

which implies that h 2 (#; %; }; ; ):

Theorem 12. Let 0: Then (#; %; }; ; ) N (%; ); where

= 1 2(1 #)(1 2})(1 + %)

(2% + # + 1) + (1 #)(1 2}): (18)

Proof. If @ 2 (#; %; }; ; ) then X1

`=1

[1 + }(` 1)][`(1 + %) + (% + #)] _`( ; )

(1 #)(1 2}) a_` 1:

We need to …nd the value of such that X1

`=1

[`(1 + %) + (% + )] _`( ; )

1 a` 1:

Thus it is su¢ cient to show that [`(1 + %) + (% + )] _`( ; )

1

[1 + }(` 1)][`(1 + %) + (% + #)] _`( ; )

(1 #)(1 2}) :

Then

1 (` + 1)(1 #)(1 2})(1 + %)

[1 + }(` 1)][`(1 + %) + (% + #)] + (1 #)(1 2}): Since

G(`) = 1 (` + 1)(1 #)(1 2})(1 + %)

[1 + }(` 1)][`(1 + %) + (% + #)] + (1 #)(1 2}) is an increasing function of `; ` 1; we obtain

G(1) = 1 2(1 #)(1 2})(1 + %) (2% + # + 1) + (1 #)(1 2}):

Theorem 13. Let @0(~) = _~¹ and

@`(~) = 1

~+ X1

`=1

(1 #)(1 2})

[1 + }(` 1)][`(1 + %) + (% + #)] _`( ; )~^`; ` 1: (19) Then @ is in the class (#; %; }; ; ) i¤ can be expressed in the form

@(~) = X1

`=0

!_`@`(~); (20)

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where !` 0 and P¹

`=0

!`= 1.

Proof. Assume that

@(~) = X1

`=0

!`@^`(~)

= 1

~+ X1

`=1

(1 #)(1 2})

[1 + }(` 1)][`(1 + %) + (% + #)] _`( ; )~^`: Then it follows that

X1

`=1

[1 + }(` 1)][`(1 + %) + (% + #)] _`( ; ) (1 #)(1 2})

(1 #)(1 2})

[1 + }(` 1)][`(1 + %) + (% + #)] _`( ; )~^`

= X1

`=1

!`= 1 !0 1

which implies that @ 2 (#; %; }; ; ):

On the other side, assume that the function @ de…ned by (1) be in the class

@ 2 (#; %; }; ; ): Then a`

(1 #)(1 2})

[1 + }(` 1)][`(1 + %) + (% + #)] _`( ; ): Setting

!_`=[1 + }(` 1)][`(1 + %) + (% + #)] _`( ; )

(1 #)(1 2}) a_`;

where

!₀= 1 X1

`=0

!_`;

@ can be expressed in the form (20), as can be shown.

Corollary 14. The extreme points of the class (#; %; }; ; ) are the functions

@⁰(~) = _~¹ and

@`(~) = 1

~+ (1 #)(1 2})

[1 + }(` 1)][`(1 + %) + (% + #)] _`( ; )~^`: (21) 5. Modified Hadamard products

Let the functions @^j(j = 1; 2) de…ned by (14). The modi…ed Hadamard product of @¹ and @² is de…ned by

(@1 @2)(~) = 1

~+ X1

`=1

a_`;1a_`;2~^`= (@2 @1)(~): (22)

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Theorem 15. Let the function @j(j = 1; 2) de…ned by (14) be in the class (#; %; }; ; ):

Then @1 @22 (#; %; }; ; ); where

' = 1 2(1 #)²(1 2})(1 + %)

(2% + # + 1)² ₁( ; ) + (1 #)²(1 2}): (23) The estimate is sharp for the functions @j(j = 1; 2) given by

@^j(~) = 1

~ + (1 #)(1 2})

(2% + # + 1) ₁( ; )~; (j = 1; 2): (24) Proof. Using the same method that Schild and Silverman [12] used earlier, we need to …nd the largest real parameter ' such that

X1

`=1

[1 + }(` 1)][`(1 + %) + (% + ')] _`( ; )

(1 ')(1 2}) a`;1a`;2 1: (25)

Since @j 2 (#; %; }; ; ); j = 1; 2; we readily see that X1

`=1

[1 + }(` 1)][`(1 + %) + (% + #] _`( ; )

(1 #)(1 2}) a_`;1 1

and X1

`=1

[1 + }(` 1)][`(1 + %) + (% + #] _`( ; )

(1 #)(1 2}) a`;2 1:

By Cauchy- Schwarz inequality, we have X1

`=1

[1 + }(` 1)][`(1 + %) + (% + #] _`( ; ) (1 #)(1 2})

pa`;1a`;2 1: (26)

Then merely demonstrating that is necessary X1

`=1

[1 + }(` 1)][`(1 + %) + (% + ')] _`( ; ) (1 ')(1 2}) a_`;1a_`;2 X1

`=1

[1 + }(` 1)][`(1 + %) + (% + #] _`( ; ) (1 #)(1 2})

pa_`;1a_`;2

or equivalently that

pa`;1a`;2

[`(1 + %) + (% + #](1 ') [`(1 + %) + (% + '](1 #):

Hence, it light of the inequality (26), then merely demonstrating that is necessary (1 #)(1 2})

[1 + }(` 1)][`(1 + %) + (% + #] _`( ; )

[`(1 + %) + (% + #](1 ')

[`(1 + %) + (% + '](1 #): (27) It follows from (27) that

' 1 (1 #)²(1 2})(1 + %)(` + 1)

[1 + }(` 1)][`(1 + %) + (% + #]² _`( ; ) + (1 #)²(1 2}):

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Now de…ning the function E(`);

E(`) = 1 (1 #)²(1 2})(1 + %)(` + 1)

[1 + }(` 1)][`(1 + %) + (% + #]² _`( ; ) + (1 #)²(1 2}): We see that E(`) is an increasing of `; ` 1: Therefore, we conclude that

' E(`) = 1 2(1 #)²(1 2})(1 + %)

(2% + # + 1)² ₁( ; ) + (1 #)²(1 2}); Hence the proof.

The following theorem is obtained using arguments close to those used in the proof of 15,

Theorem 16. Let the function @¹ de…ned by (14) be in the class (#; %; }; ; ):

Suppose also that the function @² de…ned by (14) be in the class ( ; #; %; }; ; ):

Then @¹ @²2 ( ; #; %; }; ; ); where

= 1 2(1 #)(1 )(1 2})(1 + %)

(2% + # + 1)(2% + + 1) ₁( ; ) + (1 #)(1 )(1 2}): (28) The estimate is sharp for the functions @^j(j = 1; 2) given by

@¹(~) = 1

~+ (1 #)(1 2}) (2% + # + 1) ₁( ; )~ and

@2(~) = 1

~+ (1 )(1 2}) (2% + + 1) ₁( ; )~:

Theorem 17. Let the function @^j(j = 1; 2) de…ned by (14) be in the class (#; %; }; ; ):

Then the function

h(~) = 1

~+ X1

`=1

(a²_`;1+ a²_`;2)~^` (29) belongs to the class ("; #; %; }; ; ); where

" = 1 4(1 #)²(1 2})(1 + %)

(2% + # + 1)² ₁( ; ) + 2(1 #)²(1 2}): (30) The estimate is sharp for the functions @j(j = 1; 2) given by (24).

Proof. By using Theorem 5, we obtain X1

`=1

[1 + }(` 1)][`(1 + %) + (% + #)] _`( ; ) (1 #)(1 2})

2

a²_`;1 X1

`=1

[1 + }(` 1)][`(1 + %) + (% + #)] _`( ; )

(1 #)(1 2}) a_`;1

2

1 (31)

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and

X1

`=1

[1 + }(` 1)][`(1 + %) + (% + #)] _`( ; ) (1 #)(1 2})

2

a²_`;2 X1

`=1

[1 + }(` 1)][`(1 + %) + (% + #)] _`( ; )

(1 #)(1 2}) a`;2

2

1: (32)

It follows from (31) and (32) that X1

`=1

1 2

[1 + }(` 1)][`(1 + %) + (% + #)] _`( ; ) (1 #)(1 2})

2

(a²_`;1+ a²_`;2) 1:

Therefore, we need to …nd the largest " such that [1 + }(` 1)][`(1 + %) + (% + ")] _`( ; )

(1 ")(1 2}) 1

2

[1 + }(` 1)][`(1 + %) + (% + #)] _`( ; ) (1 #)(1 2})

2

; that is

" 1 2(1 #)²(1 2})(1 + %)(` + 1)

1 + }(` 1)][`(1 + %) + (% + #)]² _`( ; ) + 2(1 #)²(1 2}): Since

G(`) = 1 2(1 #)²(1 2})(1 + %)(` + 1)

[1 + }(` 1)][`(1 + %) + (% + #)]² _`( ; ) + 2(1 #)²(1 2}) is an increasing function of `; ` 1; we obtain

" G(1) = 4(1 #)²(1 2})(1 + %)

(2% + # + 1)² ₁( ; ) + 2(1 #)²(1 2}) and hence the proof.

6. Integral operators

Theorem 18. Let the functions @ given by (1) be in the class (#; %; }; ; ):

Then the integral operator

F (~) = c Z1

0

u^c@(u~)du; 0 < u 1; c > 0 (33)

is in the class (#; %; }; ; ); where

= 1 2c(1 #)(1 + %)

(c + 2)(2% + # + 1) + c(1 #): (34) The estimate is sharp for the function @ given by (12).

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Proof. Let @ 2 (#; %; }; ; ): Then

F (~) =c Z1

0

u^c@(u~)du

=1

~ + X1

`=1

c

` + c + 1a`~^`: Thus it is enough to show that

X1

`=1

c[1 + }(` 1)][`(1 + %) + (% + )] _`( ; )

(` + c + 1)(1 )(1 2}) a` 1: (35)

Since @ 2 (#; %; }; ; ); then X1

`=1

[1 + }(` 1)][`(1 + %) + (% + #)] _`( ; )

(1 #)(1 2}) a` 1: (36)

From (35) and (36), we have

[`(1 + %) + (% + )]

(` + c + 1)(1 )

[`(1 + %) + (% + #)]

(1 #) :

Then

1 c(1 #)(` + 1)(1 + %)

(` + c + 1)[`(1 + %) + (% + #)] + c(1 #): Since

Y (`) = 1 c(1 #)(` + 1)(1 + %)

(` + c + 1)[`(1 + %) + (% + #)] + c(1 #) is an increasing function of `; ` 1; we obtain

Y (1) = 1 2c(1 #)(1 + %) (c + 2)(2% + # + 1) + c(1 #) and hence the proof.

7. Conclusion

This research has introduced a new subclass of meromorphic functions de…ned by Rapid operator and studied some basic properties of geometric function theory.

Accordingly, some results to coe¢ cient estimates, distortion properties, closure theorems, hadamard product and integral transforms have been considered, inviting further research for this …eld of study.

Author Contribution StatementsAll authors contributed equally to the design and implementation of the research, to the analysis of the results and to the writing of the manuscript.

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Declaration of Competing InterestsThe authors declare that there is no con-

‡ict of interest regarding the publication of this.

AcknowledgmentsThe authors are thankful to the editor and referee(s) for their valuable comments and suggestions which helped very much in improving the paper.

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