Research Article Univalence of a New General Integral Operator Associated with the q-hypergeometric Function

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1 Inernaional Maheaics and Maheaical Sciences Volue 23, Aricle ID , 5 pages hp://dx.doi.org/.55/23/ Research Aricle Univalence of a New General Inegral Operaor Associaed wih he q-hypergeoeric Funcion Huda Aldweby and Maslina Darus School of Maheaical Sciences,Faculy of Scienceand Technology, Universii Kebangsaan Malaysia, 436 Bangi, Selangor, Malaysia Correspondence should be addressed o Maslina Darus; aslina@uk.y Received Deceber 22; Revised 3 February 23; Acceped 7 February 23 Acadeic Edior: Shya Kalla Copyrigh 23 H. Aldweby and M. Darus. This is an open access aricle disribued under he Creaive Coons Aribuion License, which peris unresriced use, disribuion, and reproducion in any ediu, provided he original work is properly cied. Moivaed by he failiar q-hypergeoeric funcions, we inroduce a new faily of inegral operaors and obain new sufficien condiions of univalence crieria. Several corollaries and consequences of he ain resuls are also poined ou.. Inroducion Le A denoeheclassoffuncionsofhefor f () =+ c n n, c n, () which are analyic in he open uni disk U ={ C : <}, and S he class of funcions f A which are univalen in U. Le f, g A,wheref is defined by ()andg is given by g () =+ b n n, b n. (2) Then he Hadaard produc (or convoluion) f gof he funcions f and g is defined by (f g) () =+ c n b n n. (3) For coplex paraeers a i,b j, and q (i =,...,r, j =,...,s, b j C \ {,, 2,...}, q < ), we define he qhypergeoeric funcion r Φ s (a,...,a r ;b,...,b s ;q,)by (a,q) rφ s (a i ;b j ;q,) = n (a r,q) n n (4) (q, q) n (b,q) n (b s,q) n n= (r = s + ; r, s N = N {}; U), wheren denoes he se of posiive inegers and (a, q) n is he q-shifed facorial defined by, n = ; (a, q) n ={ ( a)( aq)( aq 2 ) ( aq n ), n N. (5) By using he raio es, we should noe ha, if q <,he series (4) convergesabsoluelyfor < if r=s+.for ore aheaical background of hese funcions, one ay refer o []. Corresponding o he funcion defined by (4), consider rg s (a i ;b j ;q,)= rφ s (a i ;b j ;q,). (6) Recenly, he auhors [2] defined he linear operaor M(a i,b j ; q)f : A A by where Υ n = M (a i,b j ;q)f() = r G s (a i ;b j ;q,) f() =+ Υ n c n n, (7) (a,q) n (a r,q) n, ( (q, q) n (b,q) n (b s,q) q <). (8) n

2 2 Inernaional Maheaics and Maheaical Sciences I should be rearked ha he linear operaor (7) isa generaliaion of any operaors considered earlier. For a i = q α i,b j =q β j,α i, β j C, β j =,, 2,...,(i=,...,r, j=,...,s),andq,weobainhediok-srivasavalinear operaor [3] (for r=s+), so ha i includes (as is special cases) various oher linear operaors inroduced and sudied by Ruscheweyh [4], Carlson and Shaffer [5]andheBernardi- Libera-Livingson operaors [6 8]. The q-difference operaor is defined by d q h () = h(q) h(), q=, =, (q ) li q d qh () =h (), where h () is he ordinary derivaive. For ore properies of d q see [9, ]. Lea (see [2]). Le f A;hen (9) (i) for r=,s=,anda =q,onehasm(q, ; q)f() = f(). (ii) For r =, s =, and a = q 2, one has M(q 2, ;q)f() = d q f() and li q M(q 2, ;q)f()=f (),whered q is he q-derivaive defined by (9). Definiion 2. Afuncionf A is said o be in he class B r s (a i,b j ;q;μ)if i is saisfying he condiion 2 (M (a i,b j ;q)f()) [M (a i,b j ;q)f()] 2 < μ where M(a i,b j ;q)fis he operaor defined by (7). ( U; μ<), () Noe ha B (q, ;q;μ) = B(μ), whereheclassb(μ) of analyic and univalen funcions was inroduced and sudied by Frasin and Darus []. Using he operaor M(a i,b j ; q)f()f, wenowinroduce he following new general inegral operaor. For N {}, γ,γ 2,...,γ,δ C \ {,, 2,...}, and q <, we define he inegral operaor I γk,δ(a i,b j ;q;) : A n A n by I γk,δ (a i,b j ;q;) =(δ where f k A. δ ( M (a i,b j ;q)f() f k () )/γk d) /δ, () Reark 3. I is ineresing o noe ha he inegral operaor I γk,δ(a i,b j ;q;) generalies any operaors inroduced and sudied by several auhors, for exaple, () for r = s +, a i = q α i,b j = q β j, i =,...,r, j =,...,s, q, γ k = /(α ),andδ = + (α ), where α Cand R(α) >,weobainhefollowinginegral operaor inroduced and sudied by Selvaraj and Karhikeyan [2]: F α (α,β ;) =(+(α ) (H r s (α,β )f ()) α (2) (H r s (α,β )f ()) α /(+(α )) d), where for convenience H r s (α,β )f := H(α,...,α r ;β,...,β s ; )f(), and H r s (α,β )f() = + ((α ) n (α r ) n /(β ) n (β s ) n (n )!)a n n is he Diok-Srivasava operaor [3]. (2) For r =, s =, a =q,γ k =/(α ),andδ= + (α ),weobainheinegraloperaor F,α () = (+(α ) (f ()) α (f ()) α /(+(α )) d) (3) sudied recenly by Brea e al. [3]. (3) For r =, s =, a =q,γ k =/α k,andδ=,we obain he inegral operaor F α () = ( f () ) α f ( () d )α (4) inroduced and sudied by D. Brea and N. Brea [4]. (4) For r=,s=,a =q 2,γ k = /(α ),andδ= + (α ),weobainheinegraloperaor G α () = (+(α ) (α ) (f ())α (f /(+(α )) ())α d) (5) inroduced by Selvaraj and Karhikeyan [2]. (5) For r =, s =, a =q 2,γ k =/α,andδ=,we obain he inegral operaor G α () = (f ())α (f ())α d, (6) recenly inroducedandsudied by BreaandGüney [5]. (6) For r =, s =, a =q,f = =f =f A, γ k = /(α ),andδ=α,whereα Cand R(α) >,weobain he inegral operaor G α () =(α (f ()) α /α ) d, (7) inroduced and sudied by Pescar [6]. In order o derive our ain resuls, we have o recall he following univalence crieria.

3 Inernaional Maheaics and Maheaical Sciences 3 Lea 4 (see [7, 8]). Le δ C wih Re(δ) >. Iff A saisfies f () f, (8) () for all U, hen he inegral operaor 2 Re(δ) F δ () ={δ δ f /δ () d} is in he class S. (9) Lea 5 (see [6]). Le δ Cwih Re(δ) >, c C, wih c, c =.Iff Asaisfies c 2δ +( 2δ ) f () δf, (2) () for all Uhen he inegral operaor F δ () ={δ δ f /δ () d} is in he class S. (2) Lea 6 (Generalied Schwar Lea, see [9]). (Generalied Schwar Lea) Le he funcion f be analyic in he disk U R ={: <R},wih f() < M for fixed M. Iff() has one ero wih ulipliciy order bigger ha for =,hen Equaliy can hold only if where θ is consan. f () M R, ( U R ). (22) f () =e iθ ( M R ), (23) 2. Univalence Condiions for I γk,δ(a i,b j ;q;) Theore 7. Le f k A for all k =,...,, γ k C, and M wih [(2 μ k )M+] γ. (24) k If for all,...,,f k B r s (a i,b j,q,μ k ), μ k <,and M (a i,b j ;q)f() f k () M, ( U) (25) hen he inegral operaor I γk,δ(a i,b j ;q;) defined by () is analyic and univalen in U. Proof. Fro he definiion of he operaor M(a i,b j ; q)f()f icanbeobservedha M (a i,b j ;q)f() =, ( U), (26) and for =,wehave ( M (a /γ i,b j ;q)f() f () ) ( M (a i,b j ;q)f() f () )/γ =. We define he funcion h() by he for h () = Therefore h () = (27) ( M (a i,b j ;q)f() f k () )/γk d. (28) ( M (a i,b j ;q)f() f k () )/γk. (29) Differeniaing logarihically and uliplying by on boh sides of (29) h () h () Thus we have So h () h () 2 Re(δ) = ( (M (a i,b j ;q)f() f k ()) ). γ k M (a i,b j ;q)f() f k () (3) h () h () Re(δ) 2 (M (a i,b j ;q)f() f k ()) γ. k M (a i,b j ;q)f() f k () (3) [ (M (a i,b j ;q)f() f k ()) [ γ ( +)] k M (a i,b j ;q)f() f k () ] Re(δ) 2 [ [ γ k 2 (M (a i,b j ;q)f() f k ()) ( [M (a i,b j ;q)f() f k ()] 2 M (a i,b j ;q)f() f k () +)]. ] (32)

4 4 Inernaional Maheaics and Maheaical Sciences Since M(a i,b j ; q)f()f k () M, ( U,k =,...,), and f k B r s (a i,b j,q,μ k ) for all k =,...,, hen fro he Schwar Lea and (), we obain 2 Re(δ) h () h () Re(δ) 2 2 (M (a i,b j ;q)f() f k ()) [( γ k [M (a i,b j ;q)f() f k ()] 2 M +M + )] ] γ [(2 μ k )M+], ( U) k which, in he ligh of he hypohesis (24), yields 2 Re(δ) (33) h () h, ( U). (34) () Applying Lea () forhefuncionh() we obain ha I γk,δ(a i,b j ;q;)is univalen. Taking μ k = (for all k =,...,), M =, a i = q α i,b j =q β j,q,andγ k =/(α ),δ=+(α )in Theore 7,we have he following. Corollary 8 (see [2]). Le f k A for all k =,...,and α Cwih Re (α) α 3. (35) If 2 (H r s (α,β )f k ()) (Hs r (α,β )f k ()) 2 <, ( U) (36) and for all,...,, hen he inegral operaor F α (α,β ;) defined by (2) is analyic and univalen in U. Taking μ k =(for all k =,...,), M =, r =, s =, a =q,andγ k =/(α ), δ=+(α )in Theore 7, we have he following. Corollary 9. Le f k A for all,...,and α C wih If α Re (α) 3. (37) 2 f k () (f k ()) 2 <, ( U) (38) and for all k =,...,, hen he inegral operaor F,α () defined by (3) is analyic and univalen in U. Theore. Le f k A for all k =,...,,δ, γ k C, and M wih c [(2 μ k )M+] δ γ, c C. (39) k If for all,...,, f k B r s (a i,b j,q,μ k ), μ k <,and M (a i,b j ;q)f() f k () M, ( U), (4) hen he inegral operaor I γk,δ(a i,b j ;q) defined by () is analyic and univalen in U. Proof. Fro he proof of Theore 7,wehave h () h () Thus we have = ( (M (a i,b j ;q)f() f k ()) ). γ k M (a i,b j ;q)f() f k () (4) c 2δ +( 2δ ) h () δh () c + ( 2δ ) h () δh. () (42) Fro his resul and using he proof of Theore 7 we obain c 2δ +( 2δ ) h () δh () c + δ γ [(2 μ k )M+]. k (43) Since c (/δ) (/γ k)[(2 μ k )M + ],henwehave c 2δ +( 2δ ) h () δh, ( U). (44) () Applying Lea (4) forhefuncionh() we obain ha I γk,δ(a i,b j ;q;)is univalen. Taking μ k =(for all k =,...,), r =, s =, a =q, and γ k =/(α ),δ=+(α )(α R) in Theore, we have he following. Corollary. Le f k A for all k =,...,; c C, α R, and M wih α c +( ) (2M + ), +(α ) α [, 2M + (45) 2M ]. If for all,..., 2 f k () fk 2 () <, ( U), (46) f k () M, ( U;,...,), hen he inegral operaor F,α () defined by (3) is analyic and univalen in U.

5 Inernaional Maheaics and Maheaical Sciences 5 Leing =, M =, and f =fin Corollary, we have he following. Corollary 2. Le f A, c C and α Rwih c 3 2α α, (c = ), α [, 3 (47) 2 ]. If 2 f () f 2 () <, ( U), f (), ( U), (48) hen he inegral operaor G α () defined by (7) is analyic and univalen in U. Reark 3. Many oher ineresing corollaries and resuls can be obained by specialiing he paraeers in Theore ;for exaple, see [3, 2, 2]. Acknowledgens The work presened here was parially suppored by GUP and UKM-DLP-2-5. References [] G. Gasper and M. Rahan, Basic Hypergeoeric Series,vol.35 of Encyclopedia of Maheaics and is Applicaions, Cabridge Universiy Press, Cabridge, UK, 99. [2] A. Mohaed and M. Darus, A generalied operaor involving he q-hypergeoeric funcion, Maheaici Vesnik, Available online.6.22, 2 pages. [3] J. Diok and H. M. Srivasava, Classes of analyic funcions associaed wih he generalied hypergeoeric funcion, Applied Maheaics and Copuaion,vol.3,no.,pp. 3, 999. [4] S. Ruscheweyh, New crieria for univalen funcions, Proceedings of he Aerican Maheaical Sociey, vol.49,pp.9 5, 975. [5] B.C.CarlsonandD.B.Shaffer, Sarlikeandpresarlikehypergeoeric funcions, SIAM Journal on Maheaical Analysis, vol. 5, no. 4, pp , 984. [6] S. D. Bernardi, Convex and sarlike univalen funcions, Transacions of he Aerican Maheaical Sociey,vol.35,pp , 969. [7] R. J. Libera, Soe classes of regular univalen funcions, Proceedings of he Aerican Maheaical Sociey, vol.6,pp , 965. [8] A. E. Livingson, On he radius of univalence of cerain analyic funcions, Proceedings of he Aerican Maheaical Sociey, vol. 7, pp , 966. [9] H. Exon, q-hypergeoeric Funcions and Applicaions, Ellis Horwood, Chicheser, UK, 983. [] H. A. Ghany, q-derivaive of basic hypergeoeric series wih respec o paraeers, Inernaional Maheaical Analysis,vol.3,no.33-36,pp ,29. [] B. A. Frasin and M. Darus, On cerain analyic univalen funcions, Inernaional Maheaics and Maheaical Sciences,vol.25,no.5,pp.35 3,2. [2] C. Selvaraj and K. R. Karhikeyan, Sufficien condiions for univalence of a general inegral operaor, Aca Universiais Apulensis, no. 7, pp , 29. [3] D. Brea, N. Brea, and H. M. Srivasava, An exension of he univalen condiion for a faily of inegral operaors, Applied Maheaics Leers,vol.22,no.,pp.4 44,29. [4] D. Brea and N. Brea, Two inegral operaors, Sudia Universiais Babes-Bolyai, Maheaica, Cluj-Napoca, vol.47,no. 3, pp. 3 9, 22. [5] D. Brea and H. Ö. Güney, The inegral operaor on he classes S α (b) and C α(b), Maheaical Inequaliies,vol.2,no., pp. 97, 28. [6] V. Pescar, A new generaliaion of Ahlfors s and Becker s crierion of univalence, Malaysian Maheaical Sociey Bullein, vol. 9, no. 2, pp , 996. [7] N. N. Pascu, On a univalence crierion. II, in Iineran Seinar on Funcional Equaions, Approxiaion and Convexiy (Cluj- Napoca, 985), vol.85,pp.53 54,Babeş-Bolyai Universiy, Cluj-Napoca, Roania, 985. [8] N. N. Pascu, An iproveen of Becker s univalence crierion, in Proceedings of he Coeoraive Session: Siion Soïlow, pp.43 48,UniversiyofBraşov, Braşov, Roania, 987. [9] Z. Nehari, Conforal Mapping, Dover,NewYork,NY,USA, 975. [2] D. Brea and H. Ö. Güney, On he univalence crierion of a general inegral operaor, Inequaliies and Applicaions,vol.28,AricleID7275,8pages,28. [2] G. I. Oros, G. Oros, and D. Brea, Sufficien condiions for univalenceofaninegraloperaor, Inequaliies and Applicaions,vol.28,AricleID27645,7pages,28.

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