Mapping Properties Of The General Integral Operator On The Classes R k (ρ, b) And V k (ρ, b)

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1 Applied Mahemaics E-Noes, 15(215), c ISSN Available free a mirror sies of hp:// amen/ Mapping Properies Of The General Inegral Operaor On The Classes R k (ρ, b) And V k (ρ, b) Trailokya Panigrahi, Gangadharan Murugusundaramoorhy Received 23 May 214 Absrac In his paper, we invesigae some mapping properies of wo new subclasses of analyic funcion classes R k (ρ, b) and V k (ρ, b) under generalized inegral operaor. Several (known or new) consequences of he resuls are also poined ou. 1 Inroducion Le A denoe he family of all funcions of he form: f = z + a j z j (1) which are analyic in he open uni disk U = {z C : z < 1} and saisfying he normalizaion condiions f() = and f () = 1. A funcion f A is said o be sarlike of complex order b (b C \ {}) and ype δ ( δ < 1), denoed by Sδ (b) (see [6] ) if and only if { R ( zf )} b f 1 > δ (z U). (2) A funcion f A is said o be convex of complex order b (b C \ {}) and ype δ ( δ < 1), denoed by C δ (b) (see [6] ) if and only if { R b j=2 zf } f > δ (z U). (3) For b = 1, S δ (1) = S (δ) and C δ (1) = C(δ), he classes of funcions ha are sarlike of order δ and convex of order δ in U, respecively. Clearly, C(δ) S (δ) ( δ < 1). Mahemaics Subjec Classificaions: 3C45. * Deparmen of Mahemaics, School of applied sciences, KIIT universiy, Bhubaneswar-75124, Orissa, India corresponding auhor, School of Advanced Science, VIT Universiy, Vellore-63214, Tamil Nadu, India 14

2 T. Panigrahi and G. Murugusundaramoorhy 15 Noice ha S (b) = S (b) and C (b) = C(b), he classes considered earlier by Nasr and Aouf [8] and Wiarowski [13]. Le P k (ρ) denoe he class of funcions p : U C, analyic in U saisfying he properies p() = 1 and R(p) ρ 1 ρ dθ kπ, (4) where z = re iθ U, k 2 and ρ < 1. For ρ =, we obain he class P k defined and sudied in [12]. For ρ =, k = 2, we obain he well known class P of funcions wih posiive real par and he class k = 2 gives us he class P (ρ) of funcions wih posiive real par greaer han ρ. For k > 2, he funcions in P k may no have posiive real par. I is easy o see ha p P k (ρ) if and only if here exiss h P k such ha p = (1 ρ)h + ρ. Also, from (4), we noe ha p P k (ρ) if and only if here exiss p 1, p 2 P k (ρ) such ha ( k p = ) ( k p ) p 2. 2 I is well-known ha he class P k (ρ) is a convex se (see [9]). DEFINITION. 1 A funcion f A is said o be in he class R k (ρ, b) (b C \ {}) if and only if ( zf ) b f 1 P k (ρ) (k 2, ρ < 1). (5) Noice ha R k (, 1) = R k, which is he class of funcions wih bounded radius roaion. DEFINITION. 2 A funcion f A is said o be in he class V k (ρ, b) (b C \ {}) if and only if ( zf ) b f P k (ρ) (k 2, ρ < 1). (6) We noe ha V k (, 1) V k, he class of funcions wih bounded boundary roaion firs discussed by Paaero [2]. I is clear ha f V k (ρ, b) zf R k (ρ, b). (7) Recenly, Frasin [7] inroduced he following generalized inegral operaors: Le α i, β i C for all i = 1, 2, 3,..., n, n N, γ C wih R(γ) >. Le : A n A be he inegral operaor defined by I αi,β i γ = I αi,β i γ { (f 1, f 2,..., f n ) 1 ( ) β1 ( ) βn γ γ γ 1 (f 1()) α1 f1 ()... (f n()) αn fn () d}, (8) where he power is aken as he principal one. Noice ha, he inegral operaor I αi,β i γ (f 1, f 2,..., f n ) generalizes several previously sudied operaors as follows:

3 16 Mapping Properies of he General Inegral Operaor For α i = for all i = 1, 2, 3,..., n, he inegral operaor where I,β i γ (f 1, f 2,..., f n ) = I γ (f 1, f 2,..., f n ), I γ (f 1, f 2,..., f n ) = { ( ) β1 γ γ 1 f1 ()... ( fn () ) βn d} 1 γ (9) is inroduced and sudied by Breaz and Breaz [3]. For α i = for all i = 1, 2, 3,..., n and γ = 1, he inegral operaor where F n = I,β i 1 (f 1, f 2,..., f n ) = F n, is inroduced and sudied by Breaz and Breaz [3]. ( ) β1 ( ) βn f1 () fn ()... d (1) For β i = for all i = 1, 2, 3,..., n and γ = 1, he inegral operaor where I αi, 1 (f 1, f 2,..., f n ) = F α1,α 2,...,α n, F α1,α 2,...,α n = is inroduced and sudied by Breaz e al. [5]. Recenly, Breaz and Güney [4] considered he inegral operaors F n and F α1,α 2,...,α n (f 1()) α1... (f n()) αn d (11) and obained heir properies on he classes S δ (b) and C δ(b) of sarlike and convex funcions of complex order b and ype δ. Laer on Noor e al. [1] considered he same inegral operaors and invesigaed he mapping properies for he classes V k (ρ, b) and R k (ρ, b). Moivaed by he aforemenioned work, in his paper, he auhor invesigaes some mapping properies of he classes R k (ρ, b) and V k (ρ, b) under generalized inegral operaor defined in (8) when γ = 1. The resuls obain in his paper are generalized resuls of Breaz and Güney [4] and Noor e al. [1]. 2 Main Resuls We recall he following lemma which is useful for our invesigaion:

4 T. Panigrahi and G. Murugusundaramoorhy 17 LEMMA 1 (see [11]). Le f V k (α), α < 1. Then f R k (β) where β = 1 [ (2α 1) + ] 4α 4 2 4α + 9. (12) In his secion we prove he following: THEOREM 1. Le f i, φ i R k (ρ, b) for all i = 1, 2, 3,..., n wih ρ < 1, b C \ {} and α i, β i R + for 1 i n. If hen he inegral operaor (ρ 1)n + (ρ 1) I αi, β i 1 (f 1, f 2,..., f n ) = belong o he class V k (χ, b) wih β i + 1 < 1, (13) χ = 1 + (ρ 1)n + (ρ 1) ( ) βi Π n (f i()) αi fi () d (14) β i. (15) PROOF. For he sake of simpliciy, in he proof, we shall wrie H insead of 1 (f 1, f 2,..., f n ). Differeniaing (14) wih respec o z, we obain ( ) βi H = Π n (f i) αi f. (16) z I αi, β i Le us define Clearly, φ A. Equaion (16) becomes φ = z(f i) αi. (17) H = Π n φ z n Logarihmic differeniaion of (18) yields H H = n Muliplying (19) by z and simplifying we ge b zh H = 1 n ( ) βi f. (18) z [ ( f β i f 1 ) ( φ + z φ 1 )]. (19) z b β i + φ 1 [ {β i ( )] zf b f 1 )}. (2)

5 18 Mapping Properies of he General Inegral Operaor Adding and subracing ρ on he righ hand side of (2) gives [( b zh ) ] H χ = + [( β i ( zf b [( b )) f 1 φ 1 )) ] ρ ] ρ, (21) where χ is given by (15). Taking real par of (21) and afer simplificaion, we ge [(1 R + 1 b β i [(1 R + 1 b + [(1 R + 1 b zh ) H χ] dθ ( )) zf f 1 ρ] dθ φ 1 Since, by hypohesis, f i, φ i R k (ρ, b) for 1 i n, we have and [(1 R + 1 b [(1 R + 1 b Making use of (23) and (24) in (22), we have [(1 R + 1 b )) ρ] dθ. (22) ( )) zf f 1 ρ] dθ (1 ρ)kπ (23) )) φ 1 ρ] dθ (1 ρ)kπ. (24) zh ) H χ] dθ (1 χ)kπ, (25) where χ is given by (15). Hence H V k (χ, b). Thus, he proof of Theorem 1 is compleed. Pu α i = for all i = 1, 2, 3,..., n in Theorem 1. Noice ha, in such case H = F n and φ = z which shows Therefore, from (21), i follows ha [ b zf n ] F n λ = where λ = 1 + (ρ 1) n β i. zφ i φ 1 =. [( β i ( )) ] zf b f 1 ρ,

6 T. Panigrahi and G. Murugusundaramoorhy 19 Hence we have he following Corollary 1. COROLLARY 1 (cf. [1, Theorem 2.1]). Le f R k (ρ, b) for 1 i < n wih ρ < 1 and b C \ {}. Also le β i >, 1 i n. If (ρ 1) β i + 1 < 1, hen F n V k (λ, b) wih λ = (ρ 1) n β i + 1. Nex, we ake β i =, 1 i n in Theorem 1. In his case, which implies H = Π n H = F α1,α 2,...,α n = zh H = n φ z. Therefore, n ) φ 1 = We have he following resul due o Noor e al. [1]. Π n (f i) αi, α i zf f. COROLLARY 2 (cf. [1, Theorem 2.5]). Le f V k (ρ, b) for 1 i n wih ρ < 1 and b C \ {}. Also, le α i >, 1 i n. If (ρ 1) α i + 1 < 1, hen F α1,α 2,...,α n V k (λ 1, b) wih λ 1 = (ρ 1) n α i + 1. REMARK 1. Seing k = 2 in Corollary 1, we obain he resuls of [4, Theorem 1] and [1, Corollary 2.2]. REMARK 2. Seing k = 2 in Corollary 2, we obain anoher resuls of [4, Theorem 3] and [1, Corollary 2.6]. THEOREM 2. Le f i, φ i V k (ρ, 1) for 1 i < n wih ρ < 1. Le α i, β i R +, 1 i n. If (β 1) (1 + β i ) + 1 < 1, (26) hen H V k (l, 1) wih l = 1 + (β 1) n (1 + β i) and β is given by (12). PROOF: Proceeding as Theorem 1 wih b = 1, we have [ ] R 1 + zh H l dθ [ ] β i zf R [ f β dθ + zφ ] R φ i β dθ. (27)

7 2 Mapping Properies of he General Inegral Operaor Since f i, φ i V k (ρ, l) for 1 i n, and by using Lemma 1, we have [ ] zf R f β dθ (1 β)kπ (28) and [ zφ ] R φ i β dθ (1 β)kπ, (29) where β is given by (12) wih α = ρ. Using (28) and (29) in (27), we obain [ ] R 1 + zh H l dθ (1 l)kπ. (3) Thus, H V k (l, 1) wih l = 1 + (β 1) n (1 + β i). The proof of Theorem 2 is compleed. REMARK 3. For α i = we obain he resul of ([1, Theorem 2.3]). For n = 1, α 1 =, β 1 = 1, f 1 = f in Theorem 2, we ge he following resuls due o [1]. COROLLARY 3.[1] Le f V k (ρ, 1). Then he Alexander operaor I = f() d (see [1]) belongs o he class V k (β), where β is given by (24). REMARK 4. For ρ = and k = 2 in he above Corollary 2, we have he well known resul f C() I C ( 1 2). Acknowledgemens. We record our sincere hanks o he referees for he valuable suggesions o improve our resuls. References [1] W. Alexander, Funcions which map he inerior of he uni circle upon simple regions, Ann. Mah., 17(1915), [2] S. D. Bernardi, New disorion heorems for funcions of posiive real par and applicaions o he parial sums of univalen convex funcions, Proc. Amer. Mah. Soc., 45(1974), [3] D. Breaz and N. Breaz, Two inegral operaors, Sudia Universiais Babes-Bolyai, Mahemaica, Cluj-Napoca, 3(22), [4] D. Breaz and H. Ö. Güney, The inegral operaor on he classes S α(b) and C α (b), J. Mah. Ineq., 2(28), [5] D. Breaz, S. Owa and N. Breaz, A new inegral univalen operaor, Aca Univ. Apul., 16(28),

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