Iran J Sci Technol Trans Sci https://doi.org/10.1007/s40995-017-0438-z

RESEARCH PAPER

Some Applications of Generalized Srivastava–Attiya Integral Operator S. Z. H. Bukhari1 • K. I. Noor2 • B. Malik2 Received: 22 January 2017 / Accepted: 27 November 2017 Ó Shiraz University 2017

Abstract A function analytic and locally univalent in a simply connected domain is of bounded radius rotations if its range has bounded radius rotations which is defined as the total variation of the direction angle of radial vector to the boundary curve under a complete circuit. We use Srivastava–Attiya integral operator to define some new subclasses of analytic functions which map the open unit disk to the domain related with the functions of bounded radius rotation. Some results including inclusion relations, integral preserving properties, and inverse inclusions are studied. We may relate our finding with the existing results found in the literature of the subject. Keywords Carathe´odory function  Mo¨bius function  Srivastava–Attiya integral operator  Bernardi integral operator

1 Introduction and Definitions Let HðUÞ represent the class of analytic functions f defined in the open unit disk U :¼ fz : z 2 C and jzj\1g: For a positive integer n and a 2 C, let  H½a; n :¼ f : f 2 HðUÞ and fðzÞ ¼ a þ an zn þ anþ1 znþ1

The Mo¨bius function

þ    ðz 2 UÞg:

l 0 ðzÞ ¼

We also define the class A by A :¼ f f : f 2 H½0; 1 and f 0 ð0Þ ¼ 1g:

The class of univalent functions is represented by S and it is a subclass of the class A, whereas S  ; C; K, and Q are the well-known classes of starlike, convex, close-to-convex, and quasi-convex functions, respectively. Let P denote the well-known class of Carathe´odory functions p, such that   p 2 HðUÞ; pð0Þ ¼ 1 and Re pðzÞ [ 0 ðz 2 UÞ:

ð1:1Þ

1þz ¼ 1 þ 2z þ 2z2 þ    ; 1z

ðz 2 UÞ

ð1:2Þ

maps U onto the right half-plane. If p 2 P and z ¼ reih , then 1r 1þr  RepðzÞ  j pðzÞj  1þr 1r

ðz 2 UÞ

ð1:3Þ

and & S. Z. H. Bukhari [email protected] K. I. Noor [email protected] B. Malik [email protected] 1

Department of Mathematics, Mirpur University of Science and Technology (MUST), Mirpur 10250, AJK, Pakistan

2

Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan

j p0 ð z Þ j 

2RepðzÞ 1  r2

ðz 2 UÞ:

ð1:4Þ

These inequalities are sharp. Equalities occur in (1.3) and (1.4) for suitable rotations of l0 given by (1.2). The class P k ð1Þ studied by Padmanabhan and Parvatham (1975) generalized the classes P and P k . The class P is given above, whereas the class P k introduced by Pinchuk (1971) is defined in the following. Let p be analytic in U. Then, p 2 P k , if it satisfies the conditions pð0Þ ¼ 1 and

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1 pð z Þ ¼ 2p

Z2p

ð1Þ

1 þ zeih daðhÞ 1  zeih

ðz 2 UÞ;

0

where aðhÞ : 0  h  2p is a function of bounded variation which satisfies the condition Z2p

Z2p

daðhÞ ¼ 2p

and

0

jdaðhÞj  kp:

For pj 2 P, j ¼ 1, 2, we may also write     k 1 k 1 þ p1 ð z Þ   p2 ð z Þ pð z Þ ¼ 4 2 4 2

Gd;d ðzÞ ¼ z þ

 1  X n þ d d ðn þ #  1Þ! n¼2

ðz 2 UÞ:

j RepðzÞ  1jdh  kð1  1Þp;

0

where 0  1\1 and k  2. The general Hurwitz–Lerch zeta function /, see Srivastava and Owa (1992), is given in the following: 1 z z2 þ þ þ  d d d ð 2 þ dÞ d ð 1 þ dÞ 1 X zn ¼ ; ðz 2 UÞ; d n¼0 ðn þ dÞ

/ðd; d; zÞ ¼



where d 2 CnZ , Z ¼ f1; 2; . . .g, d 2 C and for jzj ¼ 1, Red [ 1. For different values of the parameters, some applications of / are studied by Choi and Srivastava (2005), Ferreira and Lo´pez (2004), Garg et al. (2006), Lin et al. (2006), Luo and Srivastava (2005), Ruscheweyh (1975). Recently, for d 2 C and d 2 CnZ , Srivastava and Attiya (2001) introduced the operator Gd;d : A ! A, such that  1  X 1þd d n an z ; Gd;d f ðzÞ ¼ Gd;d ðzÞ  f ðzÞ ¼ z þ nþd n¼2 ðz 2 UÞ; where   1 d Gd;d ðzÞ ¼ ð1 þ dÞ /ðd; d; zÞ  d d  1  X 1þd d n ¼ z ðz 2 UÞ: nþd n¼1 Many known subclasses of A are defined using the operator Gd;d , for example, see Liu (2008). In term of convolution, the operator Ddd;# for # [  1, d 2 CnZ and d 2 C can be written as

123

ð1:5Þ

ð1Þ

Let p be analytic in U. Then, p 2 P k ð1Þ, if it satisfies the conditions pð0Þ ¼ 1 and



ðz 2 UÞ;

where

0

Z2p

Ddd;# f ðzÞ ¼ Gd;d ðzÞ  f ðzÞ ¼ z  1  X n þ d d ðn þ #  1Þ! n an z þ 1þd #!ðn  1Þ! n¼2

1þd

#!ðn  1Þ!

zn

ðz 2 UÞ:

This operator has been extensively explored in the literature of the subject. For d 2 Z, d ¼ 1 and # ¼ 0, it was investigated by Wang et al. (2010), and when d 2 Z and # ¼ 0, this is closely linked with the transformations studied by Flett, see Flett (1972). For more details, see also Ahuja (1985), Jung et al. (1993), Sa˘la˘gean (1983), Srivastava and Attiya (2007). Various generalizations of this operator are explored by Bukhari et al. (2017), Cho et al. (2010), Srivastava and Gaboury (2015), Srivastava et al. (2015), Srivastava et al. (2013). From (1.5), we have the following identities: d zðDdd;# f ðzÞÞ0 ¼ ðd þ 1ÞDdþ1 d;# f ðzÞ  dDd;# f ðzÞ ðz 2 UÞ

ð1:6Þ and zðDdd;# f ðzÞÞ0 ¼ ð# þ 1ÞDdd;#þ1 f ðzÞ  #Ddd;# f ðzÞ

ðz 2 UÞ: ð1:7Þ

For reference of (1.6) and (1.7), see Al-Shaqsi and Darus (2009). Definition 1 Let f be given by (1.1). Then, f 2 Sd ðd; #Þ, if it satisfies the condition p0 ðzÞ ¼

zðDdd;# f ðzÞÞ0 Ddd;# f ðzÞ

2P

ðz 2 UÞ;

ð1:8Þ

where d 2 C, d 2 CnZ , # [  1. We note that f 2 Sd ðd; #Þ () Ddd;# f 2 SH : In the year 1917, Loewner introduced the concept of functions with bounded boundary rotation. If f ðUÞ is a schlicht domain with a C 1 boundary curve, then the bounded boundary rotation of f ðUÞ is defined as the total variation of the boundary tangent vector argument over a complete circuit. For more general domains, the rotation is defined by a limiting process. Paatero showed that f given by (1.1) is of bounded boundary rotation, if and only if f 0 ðzÞ 6¼ 0 in U and f ðUÞ is a domain with boundary rotation at most kp. These and related classes are heavily explored in the literature of the subject. Using multiplier

Iran J Sci Technol Trans Sci

transformation, we define a new class T dk ðd; c; #Þ of analytic functions related with the functions of bounded radius rotation. Definition 2 g 2 Sd ðd; #Þ

Let f 2 A. Then, f 2 T dk ðd; c; #Þ, if for

0 d 1 zðDd;# f ðzÞÞ 1þ 1 c Ddd;# gðzÞ

! 2 Pk

ðz 2 UÞ;

ð1:9Þ

where k  2, d 2 C , c 2 Cnf0g, d 2 CnZ , and # [  1. Various known classes of analytic functions can be obtained from (1.9) for suitable choices of parameters within their specific ranges; for reference, see Al-Shaqsi and Darus (2009), Noor (1993), Noor (2000), Padmanabhan and Parvatham (1975).

This result is sharp. The proof of this lemma can easily be obtained using (1.1), (1.4), (1.6), and (1.7). Lemma 5 Let F . 2 Sd ðd; #Þ, where F . is given by (2.1). Then, f 2 Sd ðd; #Þ for jzj\r0 , where r0 ¼

1þ. pffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 þ 3 þ .2

.  0;

and this result is sharp. The proof of this lemma is simple and straight forward. Lemma 6 Miller and Mocanu (2000). Let h be convex in U and let x : U ! C, with RexðzÞ [ 0. If p is analytic in U; then pðzÞ þ xðzÞzp0 ðzÞ hðzÞ implies that

2 A Set of Preliminary Results To establish our main results, we will use the following lemmas. Lemma 1 have

Let d 2 C; d 2 CnZ and # [  1. Then, we

ðiÞ Sdþ1 ðd; #Þ  Sd ðd; #Þ; Red  0 d

and

d

ðiiÞ S ðd; # þ 1Þ  S ðd; #Þ;#  0: Lemma 2 Let f 2 Sd ðd; #Þ for d 2 C, d 2 CnZ , #  0, and z 2 U. Then, the Bernardi integral operator F . 2 Sd ðd; #Þ, where F . ð f ÞðzÞ ¼

.þ1 z.

Zz

t.1 f ðtÞdt; . [  1

ðz 2 UÞ:

0

ð2:1Þ For the proof of these lemmas, we refer Al-Shaqsi and Darus (2009). Lemma 3 Let f 2 Sd ðd; #Þ for d ¼ d1 þ id2 2 CnZ , # [  1, d 2 C and z 2 U. Then, f 2 Sdþ1 ðd; #Þ for jzj\r0 , where r0 ¼

1 þ d1 qffiffiffiffiffiffiffiffiffiffiffiffiffi ; Red ¼ d1 [ 0: 2 þ 3 þ d21

ð2:2Þ

This result is sharp. Lemma 4 Let f 2 Sd ðd; #Þ for d ¼ d1 þ id2 2 CnZ , # [  1, d 2 C and z 2 U. Then, f 2 Sd ðd; # þ 1Þ for jzj\r0 , where 1þ# pffiffiffiffiffiffiffiffiffiffiffiffiffi ; r0 ¼ 2 þ 3 þ #2

#  0:

pðzÞ hðzÞ:

3 Main Results In the following theorem, we study certain inclusions results. Theorem 1 Let d ¼ d1 þ id2 2 CnZ with Red ¼ d1  0, c 2 Cnf0g, d 2 C, # [  1, k  2 and z 2 U. Then, we have Tkdþ1 ðd; c; #Þ  Tkd ðd; c; #Þ. Proof Let f 2 T dþ1 k ðd; c; #Þ. Then, by Definition 2, there exists g 2 Sdþ1 ðd; #Þ, so that ! 0 dþ1 1 zðDd;# f ðzÞÞ H ðzÞ ¼ 1 þ ð3:1Þ  1 2 Pk : c Ddþ1 d;# gðzÞ Consider ! 0 d 1 zðDd;# f ðzÞÞ pð z Þ ¼ 1 þ 1 : c Ddd;# gðzÞ

ð3:2Þ

Since g 2 Sdþ1 ðd; #Þ, so Lemma 1 implies that g 2 Sd ðd; cÞ. Furthermore, from (1.6) and (1.8), we have fp0 ðzÞ þ dgDdd;# gðzÞ ¼ ðd þ 1ÞDdþ1 d;# gðzÞ;

ð3:3Þ

where g 2 Sd ðd; cÞ. Again using (1.6) in (3.2) and then differentiating, we write 0 0 d 0 d ðd þ 1ÞzðDdþ1 d;# f ðzÞÞ ¼ dzðDd;# f ðzÞÞ þ czp ðzÞDd;# gðzÞ

þ ½cpðzÞ  c þ 1zðDdd;# gðzÞÞ0 : ð3:4Þ Using (3.3) and (3.4) in (3.1) and then simplifying to have

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Iran J Sci Technol Trans Sci



zp0 ðzÞ ½cpðzÞ  c þ 1 1 þ1 þ p0 ð z Þ þ d c c zp0 ðzÞ ¼ pðzÞ þ 2 Pk: p0 ðzÞ þ d

HðzÞ ¼

pð z Þ ¼ ð3:5Þ

Moreover, consider that     k 1 k 1 þ p1 ð z Þ   p2 ðzÞ: pð z Þ ¼ 4 2 4 2

ð3:6Þ

z 2 U:

zp0j ðzÞ 2 P; j ¼ 1; 2: p0 ðzÞ þ #

pj ðzÞ þ

From (3.5) and (3.7), we deduce that and

in (3.12), we have      zp0 ðzÞ k 1 zp01 ðzÞ k 1  pð z Þ þ ¼ þ p1 ð z Þ þ  p 0 ðzÞ þ # 4 2 p0 ðzÞ þ # 4 2   zp02 ðzÞ

p2 ð z Þ þ : p0 ð z Þ þ # From (3.13) and (3.14), we observe that

ð3:7Þ

zp0j ðzÞ 2 P; j ¼ 1; 2 p0 ðzÞ þ d

ð3:8Þ

For Reðp0 ðzÞ þ dÞ [ 0, Lemma 2.5 and (3.8) imply that pj 2 P; j ¼ 1; 2, and z 2 U. Thus, from (3.7), we write d h f 2 T dk ðd; c; #Þ. Hence, T dþ1 k ðd; c; #Þ  T k ðd; c; #Þ. Theorem 2 Let c 2 Cnf0g, d ¼ d1 þ id2 2 CnZ , d 2 C; # [ 0, k  2 and z 2 U. Then, Tkd ðd; c; # þ 1Þ  Tkd ðd; c; #Þ. Proof Let f 2 T dk ðd; c; # þ 1Þ. Then, by Definition 2, there exists g 2 Sd ðd; # þ 1Þ, such that ! 0 d 1 zðDd;#þ1 f ðzÞÞ Q ðzÞ ¼ 1 þ ð3:9Þ  1 2 Pk: c Ddd;#þ1 gðzÞ To prove that f 2 T dk ðd; c; #Þ, we set p similar to that of (3.2). For g 2 Sd ðd; # þ 1Þ, from Lemma 1, it implies that g 2 Sd ðd; #Þ. Using the identity (1.7) in (1.8), we have fp0 ðzÞ þ #gDdd;# gðzÞ ¼ ð# þ 1ÞDdd;#þ1 gðzÞ; z 2 U: ð3:10Þ

ð# þ

¼ #zðDdd;# f ðzÞÞ0

From the operators Ddd;# and F . defined by (1.5) and (2.1), respectively, we obtain the following identity: zðDdd;# F . ðzÞÞ0 ¼ ð. þ 1ÞDdd;# f ðzÞ  gDdd;# F . ðzÞ:

þ

Theorem 3 Let d 2 C, d ¼ d1 þ id2 2 CnZ , c 2 Cnf0g, # [  1, k  2 and z 2 U. Then, for f 2 Tkd ðd; c; #Þ, F . 2 Tkd ðd; c; #Þ, .  0, where F . is given by (2.1). Proof

Consider

! 0 d 1 zðDd;# F . ð f ÞðzÞÞ pð z Þ ¼ 1 þ  1 ; .  0; c Ddd;# F . ðgÞðzÞ

On combining (3.9), (3.10), (3.11), and the following similar steps as in Theorem 1, we obtain Q ð z Þ ¼ pð z Þ þ

zp0 ðzÞ 2 Pk; p0 ðzÞ þ #

Taking

123

z 2 U:

ð3:12Þ

ð3:17Þ

where p is analytic in U and pð0Þ ¼ 1. Since g 2 Sd ðd; #Þ, so Lemma 2, we have F . ðgÞ 2 Sd ðd; #Þ. Using (3.16) in (3.17) and simplifying, we have ðDdd;# f ðzÞÞ0

¼

gðDdd;# F . ð f ÞðzÞÞ0

cp0 ðzÞDdd;# F . ðgÞðzÞ

þ

ð3:11Þ

ð3:16Þ

In the next theorems, we discuss integral preserving property of the operator F . defined by (2.1).

þ .þ1 .þ1 d ðcpðzÞ  c þ 1ÞðDd;# F . ðgÞðzÞÞ0 .þ1

0

þ czp ðzÞDdd;# gðzÞ fcpðzÞc þ 1gzðDdd;# gðzÞÞ0 :

ð3:15Þ

For Reðp0 ðzÞ þ #Þ [ 0, Lemma 6 and (3.15) imply that pj 2 P, j ¼ 1, 2. Thus, f 2 T dk ðd; c; #Þ. Hence, T dk ðd; c; # þ 1Þ  T dk ðd; c; #Þ. h

Again from (1.7) and (3.2), we can write 1ÞzðDdd;#þ1 f ðzÞÞ0

ð3:13Þ

ð3:14Þ

On combining (3.5) and (3.6), we obtain    zp0 ðzÞ k 1 zp01 ðzÞ pð z Þ þ ¼ þ p1 ðzÞ þ p0 ð z Þ þ d 4 2 p0 ð z Þ þ d    k 1 zp02 ðzÞ   p2 ðzÞ þ : 4 2 p0 ð z Þ þ d

pj ðzÞ þ

   k 1 k 1 þ p1 ð z Þ   p2 ð z Þ 4 2 4 2

: ð3:18Þ

Set p0 ðzÞ ¼

zðDdd;# F . ðgÞðzÞÞ0 Ddd;# F . ðgÞðzÞ

ð3:19Þ

:

Using (3.16) in (3.19), we can write p0 ðzÞ þ . ¼ ð. þ 1Þ

Ddd;# gðzÞ Ddd;# F . ðgÞðzÞ

:

ð3:20Þ

Iran J Sci Technol Trans Sci

Following similar arguments as in the preceding theorems, from (1.9), (3.18), (3.19), and (3.20), we can write: pð z Þ þ

zp0 ðzÞ 2 Pk; p0 ð z Þ þ .

z 2 U:

ð3:21Þ

Using  pð z Þ ¼

   k 1 k 1 þ p1 ð z Þ   p2 ð z Þ 4 2 4 2

Taking  pð z Þ ¼

for j ¼ 1; 2:

j ¼ 1; 2:

ð3:26Þ

Thus, p 2 P k for z 2 U. Hence, F . ð f Þ 2 T

d k ðd; c; #Þ.

h

In the following theorems, we include some radii problems related with the class T dk ðd; c; #Þ. Theorem 4 Let f 2 Tkd ðd; c, #Þ for c 6¼ 0, d 2 CnZ with Red ¼ d1  0, d 2 C, # [  1, k  2 and z 2 U. Then, f 2 Tkdþ1 ðd; c, #Þ for jzj\r0 , where 1 þ d1 qffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 þ 3 þ d21

d1  0:

This result is sharp.

Since g 2 Sd ðd; #Þ, so from (1.6) and (1.8), we have (3.3). Using (1.6) in (3.22) and then simplifying, we can write d ½cpðzÞ  c þ 1Ddd;# gðzÞ ¼ ðd þ 1ÞDdþ1 d;# f ðzÞ  dDd;# f ðzÞ:

This implies that ðd þ

¼dzðDdd;# f ðzÞÞ0 ½cpðzÞ 

þ zðDdd;# gðzÞÞ0 cþ1 þ czp0 ðzÞDdd;# gðzÞ: ð3:23Þ

Now, consider

where pj 2 P, j ¼ 1, 2, and z 2 U. To find that pj ðzÞ þ

zp0j ðzÞ 2P p0 ðzÞ þ d

for j ¼ 1, 2 and z 2 U, we use (1.3) and (1.4) as below.   zp0j ðzÞ Re pj ðzÞ þ p0 ð z Þ þ d ð3:27Þ ð1  r Þð1  r þ d1 þ rd1 Þ  2r :  Repj ðzÞ ð1  r Þð1  r þ d1 þ rd1 Þ The right-hand side of (3.27) is positive, if |z| \r0 , where

Proof Let f 2 T dk ðd; c; #Þ. Then, by Definition 2, there exists g 2 Sd ðd; #Þ, such that ! 0 d 1 zðDd;# f ðzÞÞ pð z Þ ¼ 1 þ  1 2 P k ; z 2 U: ð3:22Þ c Ddd;# gðzÞ

0 1ÞzðDdþ1 d;# f ðzÞÞ

   k 1 k 1 þ p1 ð z Þ   p2 ð z Þ 4 2 4 2

in (3.25), we obtain    zp0 ðzÞ k 1 zp01 ðzÞ pð z Þ þ ¼ þ p1 ðzÞ þ p0 ð z Þ þ d 4 2 p0 ð z Þ þ d    k 1 zp02 ðzÞ   p2 ð z Þ þ ; 4 2 p0 ð z Þ þ d

On applying Lemma 6, we see that pj 2 P for

r0 ¼

where H is analytic in U with Hð0Þ ¼ 1. From Lemma 3, it follows that g 2 Sdþ1 ðd; #Þ for jzj\r0 , where r0 is given by (2.2). On combining (3.3), (3.23), and (3.24), we obtain ½p0 ðzÞ þ d½cpðzÞ  c þ 1 zp0 ðzÞ 1 þ þ1 c ð p0 ð z Þ þ d Þ p0 ð z Þ þ d c 0 zp ðzÞ ¼ pð z Þ þ : ð3:25Þ p0 ð z Þ þ d

This implies that

Reðp0 ðzÞ þ .Þ [ 0;

ð3:24Þ

H ðzÞ ¼

in (3.21), we have    k 1 zp01 ðzÞ þ p1 ðzÞ þ 4 2 p0 ðzÞ þ .    k 1 zp02 ðzÞ   p2 ð z Þ þ 2 P k ; z 2 U: 4 2 p0 ð z Þ þ .

zp0j ðzÞ pj ðzÞ þ 2P p0 ðzÞ þ .

! 0 dþ1 1 zðDd;# f ðzÞÞ 1 ; H ðzÞ ¼ 1 þ c Ddþ1 d;# gðzÞ

r0 ¼

1 þ d1 qffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 þ 3 þ d21

Red ¼ d1  0

and the sharpness of r0 follows from pj ðzÞ ¼ l0 ðzÞ given by (1.2). Hence, from (3.25) and (3.26), we conclude that f 2 T dþ1 h k ðd;c ; #Þ. Theorem 5 Let f 2 Tkd ðd; c, #Þ for c 2 Cnf0g, d 2 C, #  0, k2 and d ¼ d1 þ id2 2 CnZ , d z 2 U. Then, f 2 Tk ðd; c, # þ 1Þ for jzj\r0 , where r0 ¼

1þ# pffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 þ 3 þ #2

This result is sharp. Using (1.7), (1.8), (3.22), and Lemma 4, we obtain the desired proof.

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Theorem 6 Let F . ðf Þ 2 Tkd ðd; c; #Þ for d 2 C, d ¼ d1 þ id2 2 CnZ , c 2 Cnf0g, # [  1, k  2, .  0 and z 2 U, where F . is defined by (2.1). Then, f 2 Tkd ðd; c; #Þ for jzj\r0 , where r0 ¼

1þ. pffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 þ 3 þ .2

  zp0j ðzÞ  Repj ðzÞ Re pj ðzÞ þ p0 ð z Þ þ .   ð1  r Þð1  r þ . þ rgÞ  2r : ð1  r Þð1  r þ . þ rgÞ

ð3:31Þ

The inequality (3.31) is positive, if |z| \r0 , where 1þ. pffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 þ 3 þ .2

and this result is sharp.

r0 ¼

Let F . ðf Þ 2 T dk ðd; c; #Þ. Then, ! 0 d 1 zðDd;# F . ð f ÞðzÞÞ  1 2 Pk; pð z Þ ¼ 1 þ c Ddd;# F . ðgÞðzÞ

and this value of r0 is sharp for pj ðzÞ ¼ l0 ðzÞ ¼ 1þz 1z ; j ¼ 1; 2: Hence, (3.29) and (3.30) lead to the required result. h

Proof

.  0;

z 2 U; ð3:28Þ

where F . ðgÞðzÞ 2 Sd ðd; #Þ. Using (2.1) in (3.28) and simplifying ð. þ 1ÞDdd;# f ðzÞ ¼ gDdd;# F . ð f ÞðzÞ þ ðcpðzÞ  c þ 1ÞDdd;# F . ðgÞðzÞ: On differentiating (3.28), we have (3.18). In addition, from Lemma 5, it follows that g 2 Sd ðd; #Þ for jzj\r0 , where r0 is given therein. Now, set H as in (1.9) and then use (3.18), (3.19), and (3.28) in (1.9), we have zp0 ðzÞ ðcpðzÞ  c þ 1Þp0 ðzÞ þ p0 ð z Þ þ . cðp0 ðzÞ þ .Þ .ðcpðzÞ  c þ 1Þ 1 þ þ1 cðp0 ðzÞ þ .Þ c 0 zp ðzÞ ¼ pð z Þ þ : p0 ð z Þ þ .

4 Concluding Remarks and Observations Motivated essentially by several applications of Attiya– Srivastava operator, we have introduced a new class Tkd ðd; c; #Þ of analytic functions with bounded radius rotations. For functions belonging to this general function class Tkd ðd; c; #Þ, we have derived several inclusion results, integral representations, and inverse inclusions. We have also shown how the results presented in this paper (see Theorem 1 to Theorem 6) would apply to yield the corresponding (known or new) results for a number of simpler function classes.

H ðzÞ ¼

ð3:29Þ

Taking  pð z Þ ¼

References

   k 1 k 1 þ p1 ð z Þ   p2 ð z Þ 4 2 4 2

in (3.29), we can write    zp0 ðzÞ k 1 zp01 ðzÞ pð z Þ þ ¼ þ p1 ðzÞ þ p0 ð z Þ þ . 4 2 p0 ð z Þ þ .    k 1 zp02 ðzÞ   p2 ð z Þ þ ; 4 2 p0 ð z Þ þ . ð3:30Þ where pj 2 P, j ¼ 1, 2, and z 2 U. To find that pj ðzÞ þ

zp0j ðzÞ 2P p0 ðzÞ þ .

for j ¼ 1; 2

we use (1.3) and (1.4) as below:

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and z 2 U;

Acknowledgements The authors would like to thank Worthy Vice Chancellor MUST, Mirpur, AJK, Pak, Prof. Dr. Habib-ur-Rahman (FCSP, SI) and Honorable Rector CIIT, Islamabad, Pak, Prof. Dr. SM Junaid Zaidi (HI, SI) for their untiring efforts for the promotion of research conducive environment in their relevant Institutions.

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