Stud. Univ. Babeş-Bolyai Math. 582013, No. 4, 529 537 Integal opeato defined by q-analogue of Liu-Sivastava opeato Huda Aldweby and Maslina Daus Abstact. In this pape, we shall give an application of q-analogues theoy in geometic function theoy. We intoduce an integal opeato fo meomophic functions involving the q-analogue of diffeential opeato. We also investigate seveal popeties fo this opeato. Mathematics Subject Classification 2010: 30C45. Keywods: q-analogue, meomophic function, Liu-Sivastava opeato, integal opeato. 1. Intoduction The theoy of q-analogues o q-extensions of classical fomulas and functions based on the obsevation that 1 q α lim = α, q < 1, q 1 1 q theefoe the numbe 1q α /1q is sometimes called the basic numbe [α] q. In this wok we deive q-analogue of Liu-Sivastava opeato and employ this new diffeential opeato to define an integal opeato fo meomophic functions. Let Σ denote the class of functions of the fom f = 1 z + a k z k, 1.1 k=1 which ae analytic in the punctued open unit disk U = z : z C, 0 < z < 1} = U\0}. Fo complex paametes α i, β j i = 1,...,, j = 1,..., s, α i C, β j C\0, 1, 2,...} the basic hypegeometic function o q- hypegeometic function
530 Huda Aldweby and Maslina Daus is the q-analogue of the familia hypegeometic function and it is defined as follows: 1+s k α 1, q k... α, q k ψα 1,..., α ; β 1,..., β s, q, z = 1 k q 2 z k, q, q k β 1, q k... β s, q k k=0 1.2 k with = kk 1/2, whee q 0 when > s,, s N 2 0 = N 0}, and α, q k is the q-analogue of the Pochhamme symbol α k defined by 1, k = 0; α, q k = 1 α1 αq1 αq 2... 1 αq k1, k N. It is clea that q α ; q k lim q 1 1 q k = α k. The adius of convegence ρ of the basic hypegeometic seies 1.2 fo q < 1 is given by, if < s ; ρ = 1, if = s ; 0, if > s. The basic hypegeometic seies defined by 1.2 was fist intoduced by Heine in 1846. Theefoe it is sometimes called Heine s seies. Fo moe details concening the q-theoy the eade may efe to see [1],[2]. Now fo z U, q < 1, and = s, the basic hypegeometic function defined in 1.2 takes the fom α 1 ; q k... α ; q k Φ s α 1,..., α ; β 1,..., β s, q, z = z k q; q k β 1 ; q k... β s ; q k which conveges absolutely in the open unit disk U. Coesponding to the function Φ s α 1,..., α ; β 1,..., β s, q, z, conside whee G s α 1,..., α ; β 1,..., β s, q, z = 1 z = 1 z + k=0 Φ s α 1,..., α ; β 1,..., β s, q, z k=1 α 1, q k+1... α, q k+1 q, q k+1 β 1, q k+1... β s, q k+1 z k. Next, we define the linea opeato L sα 1,..., α ; β 1,..., β s ; q : Σ Σ by L sα 1,..., α ; β 1,..., β s ; qf = G s α 1,..., α ; β 1,..., β s, q, z f sα 1, q, k = = 1 z + sα 1, q, ka k z k 1.3 k=1 α 1, q k+1... α, q k+1 q, q k+1 β 1, q k+1... β s, q k+1.
Integal opeato defined by q-analogue of Liu-Sivastava opeato 531 Fo the sake of simplicity we wite L sα 1,..., α ; β 1,..., β s ; qf = L s[α 1, q]f. Remak 1.1. i. Fo α i = q αi, β j = q βj, α i > 0, β j > 0, i = 1,..., ; j = 1,..., s, = s, q 1 the opeato L s[α 1, q]f = Hs[α 1 ]f which was investigated by Liu and Sivastava [3]. ii. Fo = 2, s = 1, α 2 = q, q 1, the opeato L 2 1[α 1, q, β 1, q]f = L[α 1 ; β 1 ]f was intoduced and studied by Liu and sivastava [4]. Futhe, we note in passing that this opeato L[α 1 ; β 1 ]f is closely elated to the Calson-Shaffe opeato L[α 1 ; β 1 ]f defined on the space of analytic univalent functions in U. iii. Fo = 1, s = 0, α 1 = λ, q 1, the opeato L 1 0[λ, q]f = D λ f = 1 z1z λ+1 fλ > 1, whee D λ is the diffeential opeato which was intoduced by Ganigi and Ualegadi [5], and then it was genealized by Yang [6]. Analogue to the integal opeato defined in [7] which involving q-hypegeometic functions on the nomalized analytic functions, we now define the following integal opeato on the space of meomophic functions in the class Σ using the diffeential opeato L s[α 1, q] defined in 1.3. Definition 1.2. Let n N, i 1, 2,..., n}, > 0. We define the integal opeato Hf 1, f 2,..., f n : Σ n Σ by Hf 1, f 2,..., f n = 1 z z 2 u L s[α 1, q]f 1 u γ1... u L s[α 1, q]f n u γn du. 1.4 0 Fo the sake of simplicity, we wite H instead of Hf 1, f 2,..., f n. We obseve that in 1.4 fo = 1, s = 0, a 1 = q, we obtain the integal opeato intoduced and studied by Mohammed and Daus [8], see also [9],[10],[11]. The following definitions intoduce subclasses of Σ which ae of meomophic stalike functions. Definition 1.3. Let a function f Σ be analytic in U. Then f is in the class Σ,sα 1, q, δ, b if and only if, f satisfies R 1 1 zl s [α 1, q]f } b L > δ, s[α 1, q]f whee L s[α 1, q]f defined in 1.3 and b C\0}, 0 δ < 1. Definition 1.4. Let a function f Σ be analytic in U. Then f is in the class Σ,sUα 1, q, α, δ, b if and only if, f satisfies R 1 1 zl s [α 1, q]f } b L > α 1 zl s [α 1, q]f s[α 1, q]f b L + δ, s[α 1, q]f whee L s[α 1, q]f defined in 1.3 and α 0, 1 δ < 1, α + δ 0, b C\0}.
532 Huda Aldweby and Maslina Daus Definition 1.5. Let a function f Σ be analytic in U. Then f is in the class Σ,sUHα 1, q, α, b if and only if, f satisfies 1 1 zl s [α 1, q]f b L 2α 2 1 s[α 1, q]f 2 < R 1 1 zl s [α 1, q]f } b L + 2α 2 1, s[α 1, q]f whee L s[α 1, q]f defined in 1.3 and α > 0, b C\0}. Fo = 1, s = 0 and α 1 = q in Definitions 1.3, 1.4 and 1.5, we obtain Σ b δ, Σ Uα, δ, b and Σ UHα, b the classes of meomophic functions, intoduced and studied by Mohammed and Daus [12]. Now, let us intoduce the following families of subclasses of meomophic functions ΣF 1 δ, b, ΣF 2 α, δ, b and ΣF 3 α, b as follows. Definition 1.6. Let a function f Σ be analytic in U. Then f is in the class ΣF 1 δ, b if and only if, f satisfies R 1 1 zzf + 3f } b zf > δ, 1.5 + 2f whee b C\0}, 0 δ < 1. Definition 1.7. Let a function f Σ be analytic in U. Then f is in the class ΣF 2 α, δ, b if and only if, f satisfies R 1 1 zzf + 3f } b zf > α 1 zzf + 3f + 2f b zf + δ, 1.6 + 2f whee α 0, 1 δ < 1, α + δ 0, b C\0}. Definition 1.8. Let a function f Σ be analytic in U. Then f is in the class ΣF 3 α, b if and only if, f satisfies 1 1 zzf + 3f b zf 2α 2 1 + 2f 2 < R 1 1 zzf + 3f } b zf + 2α 2 1, 1.7 + 2f whee α > 0, b C\0}. 2. Main esults In this section, we investigate some popeties fo the integal opeato H defined by 1.4of the subclasses given by Definitions 1.3, 1.4 and 1.5 Theoem 2.1. Fo i 1, 2,..., n}, let > 0 and f i Σ,sα 1, q, δ i, b0 δ < 1 and b C\0}. If 0 < 1 δ i 1,
Integal opeato defined by q-analogue of Liu-Sivastava opeato 533 then H is in the class ΣF 1 µ, b, µ = 1 n 1 δ i Poof. A diffeentiation of H which is defined by 1.4, we obtain z 2 H + 2zH = z L s[α 1, q]f 1 γ1... z L s[α 1, q]f n γn, 2.1 z 2 H + 4zH + 2H zl = γ s [α 1, q]f i + L s[α 1, q]f i i zl s[α 1, q]f i Then fom 2.1 and 2.2,we obtain z 2 H + 4zH + 2H z 2 H + 2zF By multiplying 2.3 with z we have z 2 H + 4zH + 2H γi zh + 2H γi That is equivalent to z zh + 3H zh + 2H [z L s[α 1, q]f 1 γ1... z L s[α 1, q]f n γn ] 2.2 = = = L s [α 1, q]f i L s[α 1, q]f i zl s [α 1, q]f i L s[α 1, q]f i zl s [α 1, q]f i L s[α 1, q]f i z.. 2.3. 2.4 Equivalently, 2.4 can be witten as 1 1 zzh + 3H } b zh +1 = 1 1 zl s [α 1, q]f i } + 2H b L +1 +1 s[α 1, q]f i Taking the eal pat of both sides of the last expession, we have R 1 1 zzh + 3H } b zh + 2H = R 1 1 zl s [α 1, q]f i } b L. s[α 1, q]f i Since f i Σ,sα 1, q, δ i, b, hence R 1 1 zzh + 3H b zh + 2H Theefoe Then H ΣF 1 µ, b, } > δ i R 1 1 zzh + 3H } b zh > 1 + 2H. 1 δ i.. µ = 1 n 1 δ i
534 Huda Aldweby and Maslina Daus Theoem 2.2. Fo i 1, 2,..., n}, let > 0 and f i Σ,sUα, δ, bα 0, 1 δ < 1, α + δ 0 and b C\0}. If 1, then H is in the class ΣF 2 α, δ, b. Poof. Since f i Σ,sUα 1, q, α, δ, b, it follows fom Definition 1.3 that R 1 1 zl s [α 1, q]f i } b L > α 1 zl s [α 1, q]f i s[α 1, q]f i b L + δ. 2.5 s[α 1, q]f i Consideing 2.2 and 2.5 we obtain R 1 1 zzh + 3H } b zh α 1 + 2H b = 1 > 1 + + = 1 δ1 R 1 1 zl s [α 1, q]f i b L s[α 1, q]f i α α 1 zl s [α 1, q]f i b L s[α 1, q]f i α 0. This completes the poof. zzh + 3H zh + 2H 1 b } zl s [α 1, q]f i L s[α 1, q]f i } + δ δ. δ 1 zl γ s [α 1, q]f i i b L δ s[α 1, q]f i Theoem 2.3. Fo i 1, 2,..., n}, let > 0 and f i Σ UHα, b α > 0 and b C\0}. If 1, then H is in the class ΣF 3 α, b. Poof. Since f i Σ,sUHα 1, q, α, b, it follows fom Definition 1.4 that 2 R 1 1 zl s [α 1, q]f i } b L + 2α 2 1 s[α 1, q]f i 1 1 zl s [α 1, q]f i b L 2α 2 1 s[α 1, q]f i > 0. 2.6
Integal opeato defined by q-analogue of Liu-Sivastava opeato 535 Consideing 2.2 and 2.6, we obtain 2 R 1 1 zzh + 3H b zh + 2H 1 1 b = R 2 [ = 2 2 = 2 + 2 = 2 1 + 1 1 } 1 zl γ s [α 1, q]f i i b L s[α 1, q]f i n 1 zzh + 3H zh + 2H 1 b R 1 zl s [α 1, q]f i b L s[α 1, q]f i n 1 1 b R 1 1 zl s [α 1, q]f i b L s[α 1, q]f i [ 1 1 b zl s [α 1, q]f i L s[α 1, q]f i + 2α 2 1 + 2 [1 2α 2 1] 1 2 1 1 1 b + + 2α 2 1 2α 2 1 2.7 ]} + 2α 2 1 zl s [α 1, q]f i L s[α 1, q]f i + 2α 2 1 zl s [α 1, q]f i L s[α 1, q]f i + 2α 2 1 + 2 zl s [α 1, q]f i L s[α 1, q]f i } 2 2α 2 1 2α 2 1 + 2α 2 1 2α 2 1 + 2α 2 1 ] 2α 2 1 R 1 1 zl s [α 1, q]f i b L s[α 1, q]f i [ 1 1 b zl s [α 1, q]f i L s[α 1, q]f i } 2α 2 1] R 1 1 zl s [α 1, q]f i b L s[α 1, q]f i 2α 2 1 12α 21 1 }
536 Huda Aldweby and Maslina Daus = R [ 2 1 1 zl s [α 1, q]f i b L s[α 1, q]f i ] + 2α 2 1 1 1 zl s [α 1, q]f i b L 2α } 2 1 s[α 1, q]f i + 2 1 + 2α 2 1 2α 2 1 1 2α 2 1 1 > [ 2 + 2α 2 1 1 2α 2 1 ] 1 > 1 min 2 11 + 4α, 2 } 0. This completes the poof. Acknowledgements. The wok is suppoted by GUP-2013-004. Refeences [1] Gaspe, G., Rahman, M., Basic Hypegeometic Seies, 35, Cambidge Univesity Pess, Cambidge, 1990. [2] Exton, H., q-hypegeometic functions and applications, Ellis Howood Limited, Chicheste, 1983. [3] Liu, J., Sivastava, H.M., Classes of meomophically multivalent functions associated with the genealized hypegeometic function, Mathematical and Compute Modelling, 392004, no. 1, 21-34. [4] Liu, J., Sivastava, H.M., A linea opeato and associated families of meomophically multivalent functions, Jounal of Mathematical Analysis and Applications, 2592001, no. 2, 566-581. [5] Ganigi, M.R., Ualegaddi, B.A., New citeia fo meomophic univalent functions, Bulletin Mathématique de la Société des Sciences Mathématiques de la République Socialiste de Roumanie. Nouvelle Séie, 331989, no. 1, 9-13. [6] Yang, D., On a class of meomophic stalike multivalent functions, Bulletin of the Institute of Mathematics. Academia Sinica, 241996, no. 2, 151-157. [7] Aldweby, H., Daus, M., Univalence of a New Geneal Integal Opeato Associated with the q-hypegeometic Function, Intenational Jounal of Mathematics and Mathematical Sciences, 2013, Aticle ID 769537, 5 pages. [8] Mohammed, A., Daus, M., A new integal opeatofo meomophic functions, Acta Univesitatis Apulensis, 2010, no. 24, 231-238. [9] Mohammed, A., Daus, M., Stalikeness popeties of a new integal opeato fo meomophic functions, Jounal of Applied Mathematics, 2011, Aticle ID 804150, 8 pages. [10] Mohammed, A., Daus, M., Some popeties of cetain integal opeatos on new subclasses of analytic functions with complex ode, Jounal of Applied Mathematics, 2012, Aticle ID 161436, 9 pages.
Integal opeato defined by q-analogue of Liu-Sivastava opeato 537 [11] Mohammed, A., Daus, M., The ode of stalikeness of new p-valent meomophic functions, Intenational Jounal of Mathematical Analysis, 62012, no. 27, 1329-1340. [12] Mohammed, A., Daus, M., Integal opeatos on new families of meomophic functions of complex ode, Jounal of Inequalities and Applications, 1212012, 12 pages. Huda Aldweby Univesiti Kebangsaan Malaysia School of Mathematical Sciences 43600, Bangi, Selango, Malaysia e-mail: h.aldweby@yahoo.com Maslina Daus Univesiti Kebangsaan Malaysia School of Mathematical Sciences 43600, Bangi, Selango, Malaysia e-mail: maslina@ukm.my