13. Inclusion properties of certain classes of meromorphic spiral-like functions of complex order associated with generalized hypergeometric function

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ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 3 Issue 3(2011), Pages 115-121.

INCLUSION PROPERTIES OF CERTAIN CLASSES OF MEROMORPHIC SPIRAL-LIKE FUNCTIONS OF COMPLEX

ORDER ASSOICATED WITH THE GENERALIZED HYPERGEOMETRIC FUNCTION

( C O M M U N IC AT E D B Y S H IG E Y O S H I O WA )

ALI MUHAMMAD

Abstract. The purpose of the present paper is to introduce new classes of meromorphic spiral-like functions de…ned by using a meromorphic analogue of the Choi-Saigo-Srivastava operator for the generalized hypergeometric function and investigate a number of inclusion relationships of these classes.

1. Introduction

LetM denote the class of functions of the form

f(z) =1

z +

1 X

k=0

ak zk; (1.1)

which are analytic in the punctured unit diskE =_fz : 0<_jz_j<1_g=E_nf0_g:

If f and g are analytic in E = E _{[ f}0_g; we say that f is subordinate to g;

written f g or f(z) g(z); if there exists a Schwarz functionwin E such that

f(z) =g(w(z)):

LetP be the class of all functions which are analytic and univalent inE and for which (E)is convex with (0) = 1and Re_f (z_g>0 (z₂E):

For a complex parameters 1; ::: q and 1; ::: s ( j 2CnZ0 =f0; 1; 2; :::g;

j = 1; :::s); we now de…ne the generalized hypergeometric function[16;17] as fol-lows:

qFs( 1; ::: q; 1; ::: s) =

1 X

k=0

( 1)k:::( q)k

( ₁)k:::( s)k k!

zk; (1.2)

2000Mathematics Subject Classi…cation. 30C45, 30C50.

Key words and phrases. Meromorphic functions; Spiral-like functions of complex order; Hadamard product; Di¤erential subordination; Choi-Saigo-Srivastava operator.

c 2011 Universiteti i Prishtinës, Prishtinë, Kosovë. Submitted March 17, 2011. Accepted June 2, 2011.

Dedicated to Mr. and Mrs. Ibrahim Amodu Nigeria.

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where(q s+ 1;s₂N[ f0_g;N=_f1;2; :::_g) and(v)k is the Pochhammer symbol

(or shifted factorial) de…ned in (terms of the Gamma function) by

(v)k =

(v+k) (v) =

1 ifk= 0 andv₂Cnf0_g

v(v+ 1):::(v+k 1) ifk₂Nandv₂C:

Corresponding to a function

F( 1; ::: q; 1; ::: s;z) =z 1

qFs( 1; ::: q; 1; ::: s;z): (1.3)

Liu and Srivastava[11]consider a linear operatorH( 1; ::: q; 1; ::: s) :M !M

de…ned by the following Hadamard product(or convolution):

H( 1; :: q; 1; :: s)f(z) =h( 1; ::: q; 1; ::: s;z) f(z):

We note that the linear operatorH( 1; :: q; 1; :: s)was motivated essentially by

Dzoik and Srivastava[4]:Some interesting developments with the generalized hyper-geometric function were considered recently by Dzoik and Srivastava[5;6]and Liu and Srivastava[9;10]:Corresponding to the functionh( 1; ::: q; 1; ::: s;z)de…ned

by(1:3); we introduce a functionh ( 1; ::: q; 1; ::: s;z)given by

h( 1; ::: q; 1; ::: s;z) h ( 1; ::: q; 1; ::: s;z) =

1

z(1 z) ( >0): (1.5) Analogous toH( 1; :: q; 1; :: s)de…ned by(1:4);we now de…ne the linear operator

H ( 1; :: q; 1; :: s)onM as follows:

H ( 1; :: q; 1; :: s)f(z) =h ( 1; ::: q; 1; ::: s;z) f(z); (1.6)

where i; j 2CnZ0;i= 1; ::q;j= 1; ::s; >0;z2E ; f 2M:

For convenience, we write

H ;q;s( 1) =H ( 1; :: q; 1; :: s):

It is easily veri…ed from the de…nition(1:5)and(1:6) that

z(H ;q;s( 1+ 1)f(z))

0

= 1H ;q;s( 1)f(z) ( 1+ 1)H ;q;s( 1+ 1)f(z); (1.7)

and

z(H ;q;s( 1)f(z))

0

= H +1;q;s( 1)f(z) ( + 1)H ;q;s( 1)f(z): (1.8)

We note that the operatorH ;q;s( 1)is closely related to the Choi-Saigo-Srivastava

operator[3]for analytic functions, which includes the integral operator studied by Liu[8]and Noor et al [13;15]:The interested readers are refered to the work done by the authors[1;2;14]:

De…nition 1.1. Using the subordination principle between two analytic func-tions, we introduce the subclassesM S_b( (z)); M C_b( (z))andM K_b;c; ( (z); (z)) of the classM as follows:

M S_b( (z)) = f(z)₂M : 1 + e

i

bcos

zf0(z)

f(z) 1 (z) inE

M C_b( (z)) = f(z)₂M : 1 + e

i

bcos

(zf0₍_z₎₎0

f0₍_z₎ 1 (z) in E

M K_b;c; ( (z); (z)) = f(z)₂M : 1 + e

i

bcos

zf0₍_z₎

g(z) 1 (z), ; withg(z)₂M S_b( (z))in E and ; ₂R:_{j j}< ₂; _{j j}< ₂; b; c₆= 0with b:c₂C

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Now by using the operator (H ;q;s( 1);we introduce a new subclasses of

mero-morphic functions.

M S_{b; ;q;s;} ₁( (z)) = _ff(z)₂M : H ;q;s( 1)f(z)2M Sb( (z))g (1.9)

M C_{b; ;q;s;} ₁( (z)) = _ff(z)₂M : H ;q;s( 1)f(z)2M Cb( (z))g (1.10)

M K_{b;c; ;q;s;;}; ₁( (z); (z)) = nf(z)₂M :H ;q;s( 1)f(z)2M K_b;c; ( (z); (z)) o

;

(1.11) where ; ₂R:_{j j}< ₂; _{j j}< ₂; b; c₆= 0 withb:c₂Cand (z); (z)₂P:From (1:9) and(1:10);it is clear that

f(z)₂M C_{b; ;q;s;} ₁( (z))₍₎ zf0(z)₂M S_{b; ;q;s;} ₁( (z)): (1.12)

2. Preliminary Results

To establish our main results we need the following Lemmas.

Lemma 2.1[7]:Let be convex univalent inE with (0) = 1 andRe_f (z) +

t)>0 ( ,t₂C): Ifpis analytic inE withp(0) = 1;then

p(z) + z p 0₍_z₎

p(z) +t (z) (z2E),)p(z) (z):

Lemma 2.2 [12]:Let (z)₂P be convex univalent inE and!(z)be analytic inE withRe_f!(z)_g 0: If pis analytic inE withp(0) = (0);then

p(z) +!(z)zp0(z) (z) (z₂E) =₎p(z) (z):

3. Main Results

Theorem 3.1. Let ₂R;where _{j j}< ₂ and letb=b1+ib26= 0;tan = b_b2₁;

(z)₂P forz₂E ( ; 1>0):Then

M S_b; ₊₁_;q;s; ₁( (z)) M S_{b; ;q;s;} ₁( (z)) M S_{b; ;q;s;} ₁₊₁( (z));

forIm (z)<(Re (z) 1) cot( v):

Proof. To prove the …rst part of Theorem 3.1, letf ₂M S_b; ₊₁_;q;s; ₁( (z))and set

p(z) = 1

bcos e

i z(H ;q;s( 1)f(z))0 H ;q;s( 1)f(z)

(1 b) cos isin : (3.1)

Thenp(z)is analytic inEwithp(0) = 1:Applying(1:8)in(3:1)and with a simple computations, we have for >0

1 + e

i

bcos

z(H +1;q;s( 1)f(z))0 H +1;q;s( 1)f(z)

1 =p(z)+ zp 0₍_z₎

e i _b_{cos (}_p₍_z₎ _{1) +} _{+ 1} (z):

(3.2) SinceRe_f e i bcos ( (z) 1) + + 1)>0forIm (z)<(Re (z) 1) cot( v) and wheretan = b2

b1;so by Lemma 2.1 and(3:2);we havep(z) (z):This proves

that

M S_b; ₊₁_;q;s; ₁( (z)) M S_{b; ;q;s;} ₁( (z)): (3.3) To prove the second part of Theorem 3.1, we consider

p(z) = 1

bcos e

i z(H ;q;s( 1+ 1)f(z))0 H ;q;s( 1+ 1)f(z)

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Thenp(z)is analytic inEwithp(0) = 1:Applying(1:7)in(3:1)and with a simple computation, we have for 1>0

1 + e

i

bcos

z(H ;q;s( 1)f(z))0 H ;q;s( 1)f(z)

1 =p(z)+ zp 0₍_z₎

e i _b_{cos (}_p₍_z₎ _{1) +}

1+ 1

(z):

(3.5) SinceRe_f e i _b_{cos ( (}_z₎ _{1) +}

1+ 1)>0forIm (z)<(Re (z) 1) cot( v)

and where tan = b2

b1; so by Lemma 2.1 and (3:5); we have p(z) (z): This

complete the proof of second inclusion.

Theorem 3.2. Let ₂R;where _{j j}< ₂ and letb=b1+ib26= 0;tan = b_b2₁;

(z)₂P forz₂E ( ; 1>0):Then

M C_b; ₊₁_;q;s; ₁( (z)) M C_{b; ;q;s;} ₁( (z)) M C_{b; ;q;s;} ₁₊₁( (z)):

forIm (z)<(Re (z) 1) cot( v); z₂E:

Proof. The proof follows from Theorem 3.1 and(1:12):

Taking

(z) = 1 +Az

1 +Bz ( 1< B < A 1;z2E)

Corollary 3.3. Let ₂R;where _{j j}< ₂ and letb=b1+ib26= 0;tan = b_b2₁; 1+A

1+B <minf + 1=e

i _b_cos _;

1+ 1=e i bcos g; 1< B < A 1:Then

M S_b; ₊₁_;q;s; ₁(A; B)) M S_{b; ;q;s;} ₁(A; B) M S_{b; ;q;s;} ₁₊₁(A; B);

and

M C_b; ₊₁_;q;s; ₁(A; B) M C_{b; ;q;s;} ₁(A; B) M C_{b; ;q;s;} ₁₊₁(A; B):

Next, by using Lemma 2.2, we obtain the following Inclusion relation for the class of meromorphically close to convex functions.

Theorem 3.4. Let ; ₂R; where_{j j}< ₂; _{j j}< ₂ and letb=b1+ib26= 0;

tan =b2

b1; (z); (z)2P forz2E:Then

M K_b;c;; ₊₁_;q;s;; ₁( (z); (z)) M K_{b;c; ;q;s;;}; ₁( (z); (z)) M K_{b;c; ;q;s;;}; ₁₊₁( (z); (z));

forIm (z)<Re(Re (z) 1) cot( v);Im(q(z)<(Req(z) 1) cot( v)(z₂E);

( ; 1>0):

Proof. To prove the …rst inclusion of Theorem 3.4, letf ₂M K_b;c;; ₊₁_;q;s;; ₁( (z); (z)):

Then from the de…nition of M K_b;c;; ₊₁_;q;s;; ₁( (z); (z)); there existsa function

g₂M S_b; ₊₁_;q;s; ₁( (z))) such that

1

ccos e

i z(H +1;q;s( 1)f(z))0 H +1;q;s( 1)g(z)

(1 c) cos isin (z):(z₂E):

(3.6) Now let

p(z) = 1

ccos e

i z(H ;q;s( 1)f(z))0 H ;q;s( 1)g(z)

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wherep(z)is analytic inE withp(0) = 1:Using(1:8);we obtain that 1

ccos e

i z(H +1;q;s( 1)f(z))0 H +1;q;s( 1)g(z)

(1 c) cos isin

= 1

ccos

0 @ei

z(H ;q;s( 1)( zf(z))0

H ;q;s( 1)g(z) + ( + 1)

z(H ;q;s( 1)( zf(z))

H ;q;s( 1)g(z) (H ;q;s( 1)g(z))0

H ;q;s( 1)g(z) + + 1

(1 c) cos isin

1 A:

(3.8) Sinceg(z)₂M S_b; ₊₁_;q;s; ₁( (z)) M S_{b; ;q;s;} ₁( (z));by Theorem 3.1, we set

q(z) = 1

bcos e

i z(H ;q;s( 1)f(z))0 H ;q;s( 1)g(z)

(1 b) cos isin ; (3.9)

whereq inEwith assumption ₂P:Then , by virtue of(3:7);(3:8)and(3:9);

we obtain that 1

ccos e

i z(H +1;q;s( 1)f(z))0 H +1;q;s( 1)g(z)

(1 c) cos isin

= p(z) + zp0(z)

e i ₍_q₍_z₎ _{1) +} _{+ 1} (z) (z2E): (3.10)

Sinceq and >0 in E withRe( e i (q(z) 1) + + 1)>0 forIm (z)<

Re(Re (z) 1) cot( v);Im(q(z)<(Req(z) 1) cot( v):Hence, by taking

!(z) = 1

e i ₍_q₍_z₎ _{1) +} _{+ 1};

in (3:10); and applying Lemma2.2, we can show that p(z) (z) in E; so that

f(z)₂M K_{b;c; ;q;s;;};

1( (z); (z)):Moreover, we have the second inclusion by using

the similar arguments to those detailed above with (1:7): Therefore we complete the proof of the Theorem 3.4.

Inclusion properties involving the Integral operaton F

Consider the operatorF ;de…ned by

F (f)(z) =

z +1

z

Z

0

t f(t)dt (f ₂M; >0): (3.11)

From the de…nition ofF de…ned by(3:11); we observe that

z(H ;q;s( 1)F f(z))0= H ;q;s( 1)f(z) ( + 1)H ;q;s( 1)F f(z): (3.12) Theorem 3.5. Let ₂R; where _{j j} < ₂ and let b = b1+ib2 6= 0; (z) 2 P for z ₂ E ( ; 1 > 0): Then for f(z) 2 M Sb; ;q;s; 1( (z)); then F (f)(z) 2 M S_{b; ;q;s;} ₁( (z));forIm (z)<(Re (z) 1) cot( v);wheretanv= b2

b1; z2E: Proof. Consider

p(z) = 1

bcos e

i z(H ;q;s( 1)F (f)(z))0 H ;q;s( 1)F (f)(z)

(1 b) cos isin ; (3.13)

where p(z)is analytic in E withp(0) = 1: Using(3:12) in(3:13)and after simple computation we have

p(z) + zp0(z)

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ForIm (z)<(Re (z) 1) cot( v);wheretanv=b2

b1;we have

Re( e i bcos (p(z) 1) + )>0:

Thus, by Lemma 2.1 yeildsp(z) (z):Hence we have the desired proof.

Next, we derive an inclusion property involvingF which is obtaind by applying (1:8) and Theorem 3.5.

Theorem 3.6. Let ₂R; where _{j j} < ₂ and let b = b1+ib2 6= 0; (z) 2 P for z ₂ E ( ; 1 > 0): Then for f(z) 2 M Cb; ;q;s; 1( (z)); then F (f)(z) 2 M C_{b; ;q;s;} ₁( (z));forIm (z)<(Re (z) 1) cot( v);wheretanv= b2

b1; z2E:

Finally, we obtain Theorem 3.7 below by using the same lines of proof as we used in the proof of Theorem3.4.

Theorem 3.7. Let ; ₂R;where_{j j}< ₂;_{j j}< ₂ and letb=b1+ib26= 0;

tan =b2

b1; (z); (z)2Pforz2E( ; 1>0):Iff 2M K

;

b;c; +1;q;s;; 1( (z); (z));

ThenF (f)(z)₂M K_b;c;; ₊₁_;q;s;; ₁( (z); (z)) ( >0)for

Im (z)<(Re (z) 1) cot( v); Imq(z)<(Req(z) 1) cot( v)andq(z) (z); z₂E:

Acknowledgments. The authors would like to thank S. Owa for his comments that helped us improve this article.

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Ali Muhammad, Departement of Basic Sciences, University of Engineering and Tech-nology Peshawar, Pakistan