Integral operator defined by q-analogue of Liu-Srivastava operator

Transcription

1 Stud. Univ. Babeş-Bolyai Math , No. 4, 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

2 530 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 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 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.

3 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 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}.

4 532 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, f 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, f 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 f 2 < R 1 1 zzf + 3f } b zf + 2α 2 1, f 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,

5 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 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 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

6 534 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 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 >

7 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 = = } 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 [1 2α 2 1] b + + 2α 2 1 2α ]} + 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 zl s [α 1, q]f i L s[α 1, q]f i } 2 2α 2 1 2α α 2 1 2α α 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 α 21 1 }

8 536 Huda Aldweby and Maslina Daus = R [ zl s [α 1, q]f i b L s[α 1, q]f i ] + 2α zl s [α 1, q]f i b L 2α } 2 1 s[α 1, q]f i α 2 1 2α α > [ 2 + 2α α 2 1 ] 1 > 1 min α, 2 } 0. This completes the poof. Acknowledgements. The wok is suppoted by GUP Refeences [1] Gaspe, G., Rahman, M., Basic Hypegeometic Seies, 35, Cambidge Univesity Pess, Cambidge, [2] Exton, H., q-hypegeometic functions and applications, Ellis Howood Limited, Chicheste, [3] Liu, J., Sivastava, H.M., Classes of meomophically multivalent functions associated with the genealized hypegeometic function, Mathematical and Compute Modelling, , no. 1, [4] Liu, J., Sivastava, H.M., A linea opeato and associated families of meomophically multivalent functions, Jounal of Mathematical Analysis and Applications, , no. 2, [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, , no. 1, [6] Yang, D., On a class of meomophic stalike multivalent functions, Bulletin of the Institute of Mathematics. Academia Sinica, , no. 2, [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 , 5 pages. [8] Mohammed, A., Daus, M., A new integal opeatofo meomophic functions, Acta Univesitatis Apulensis, 2010, no. 24, [9] Mohammed, A., Daus, M., Stalikeness popeties of a new integal opeato fo meomophic functions, Jounal of Applied Mathematics, 2011, Aticle ID , 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 , 9 pages.

9 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, [12] Mohammed, A., Daus, M., Integal opeatos on new families of meomophic functions of complex ode, Jounal of Inequalities and Applications, , 12 pages. Huda Aldweby Univesiti Kebangsaan Malaysia School of Mathematical Sciences 43600, Bangi, Selango, Malaysia Maslina Daus Univesiti Kebangsaan Malaysia School of Mathematical Sciences 43600, Bangi, Selango, Malaysia