On Certain Properties of Neighborhoods of. Dziok-Srivastava Differential Operator

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1 International Matheatical Foru, Vol. 6, 20, no. 65, On Certain Proerties of Neighborhoods of -Valent Functions Involving a Generalized Dziok-Srivastava Differential Oerator Hesa Mahzoon Deartent of Matheatics Firoozkooh Branch,Islaic Azad university Firoozkooh, Iran ahzoon hesa@yahoo.co Abstract. In this aer, we introduce a generalized Dziok - Srivastava differential oerator H, q,s α,β ) and using this oerator, the new subclasses H, q,sα,β,n,,b), q,s α,β,n,,b; μ), Hq,s,,α α,β,n,,b) and L,,α q,s α,β,n,,b; μ) of the class L, of - valent functions denoted by A n) are defined. Further for functions belonging to these classes, certain roerties of neighborhoods are studied. Matheatics Subject Classification: 30C45

2 3236 H. Mahzoon Keywords: Analytic functions, Coefficient bounds n, δ) - neighborhood, - valent functions and generalized Dziok - Srivastava differential oerator. INTRODUCTION Let A n) be the class of noralized functions f of the for.) fz) =z + a k z k, n, N), which are analytic and -valent in the oen unit disc U = {z C : z < }. Let T n) be the subclass of A n) consisting functions f of the for.2) fz) =z a k z k, a k 0, n, N), which are - valent in U. The Hadaard roduct of two ower series fz) =z + is defined as f g)z) =z + a k z k and gz) =z + a k b k z k. b k z k Definition.. For α i C i =, 2,..., q) and β i C \{0,, 2,..., } i =, 2,..., s), q s +, q, s N 0 and R 0) the oerator H, q,s α,..., α q ; β,..., β s ):A A, A ) A ) is defined by where, H, q,s α,..., α q ; β,..., β s )= z q F sα,..., α q ; β,..., β s ; z) ] D fz) D fz) = )fz)+ zf z) and qf s α,..., α q ; β,..., β s ; z) = α ) k... α q ) k z k, z U) β ) k... β s ) k k! k=0

3 Generalized Dziok-Srivastava oerator 3237 where, q s +; q, s N 0 = {0,, 2,...} and x) k is the Pochhaer sybol defined by for k =0 x) k = xx +)...x + k ) for k =, 2, 3,... For notational silicity, we write H, q,s α,..., α q ; β,..., β s )=H, q,s α,β ). In articular, we ut H, q,s α,β )=H q,s α,β ). Reark.2. We observe that for we have.3) H, q,s α,β )fz) =z + k=+ fz) =z + k=+ a k z k, ) ] k + α ) k... α q ) k β ) k... β s ) k k )! a kz k. Reark.3. It is easy to observe that for =0we obtain the Dziok - Srivastava differential oerator ]. Definition.4. A function f T n) is said to be in the class H, q,sα,β,n,,b) if.4) b z H, q,s α,β )fz) ) +) H, q,s α,β )fz)) ) ) ) <, where, N, N 0, α > 0, β > 0, 0, >,b C \{0} and z U.

4 3238 H. Mahzoon Definition.5. A function f T n) is said to be in the class L, q,s α,β,n,,b; μ) if.5) H, μ) b q,s α,β )fz) z ) ) + μ H, q,s α,β )fz) ) +) ) ] <, where, N, N 0, α > 0, β > 0, 0, >, μ 0, b C \{0} and z U. For any function f T n) and δ 0, the n, δ) - neighborhood of f is defined as,.6) { N n,δ f) = g T n) :gz) =z b k z k and } k a k b k δ. For the function hz) =z, N) we have, {.7) N n,δ h) = g T n) :gz) =z b k z k and } k b k δ. The concet of neighborhoods was first introduced by Goodan 3] and then generalized by Ruscheweyh 7]. 2. Coefficient bounds In this section, we deterine the coefficient inequalities for functions to be in the subclasses H, q,sα,β,n,,b) and L, q,sα,β,n,,b; μ). Theore 2.. Let f T n). Then, f H, q,s α,β,n,,b) if and only if 2.) ) ] ) k + α ) k...α q ) k β ) k...β s ) k k )! k + b ) a k b.

5 Generalized Dziok-Srivastava oerator 3239 Proof. Let f H, q,s α,β,n,,b). Then, by.4) and.5) we can write, 2.2) R ) ) ] k + α ) k...α q ) k ) k + ) k β ) k...β s ) k k )! ] α ) k...α q ) k β ) k...β s ) k k )! k)a k z k ) > b. k a k z k Taking z = r, 0 r<) in 2.2), we see that the exression in the denoinator on the Left Hand Side of 2.2), is ositive for r = 0 and also for all r, 0 r<. Hence, by letting r through real values, exression 2.2) yields the desired condition 2.). Conversely, by alying the hyothesis 2.) and letting z =, we obtain, z Hq,s, α,β )fz) ) +) ) Hq,s, α,β )fz)) ) = ) z b ) ] k + α ) k...α q ) k β ) k...β s ) k ) ) k + k + + ) ) k ) k )! ] α ) k...α q ) k β ) k...β s ) k k )! ] α ) k...α q ) k β ) k...β s ) k ] α ) k...α q ) k β ) k...β s ) k k )! ) k k)a k z k ) k a k z k ] k )a k k )! ) k a k = b. Hence, by the axiu odulus theore, we have f H, q,s α,β,n,,b). Thus the roof is colete. On siilar lines, we can rove the following Theore.

6 3240 H. Mahzoon Theore 2.2. A function fz) L, q,s α,β,n,,b; μ) if and only if ) ] k + α ) k...α q ) k β ) k...β s ) k k )! k ) μk ) + ] a k 2.3) b ) +! )]. 3. Inclusion relationshis involving n, δ) - neighborhoods In this section, we rove certain inclusion relationshis for the functions belonging to the classes H, q,s α,β,n,,b) and L, q,s α,β,n,,b; μ). Theore 3.. If 3.) δ = n + b ) then H, q,s α,β,n,,b) N n,δ h). n + ) b ) + n ) α ) n...α q ) n β ) n...β s ) n ), > b ), n + Proof. Let f H, q,sα,β,n,,b). By Theore 2., we have, n + b ) + n ) ) α ) n...α q ) n n + β ) n...β s ) n ) a k b which ilies, 3.2) a k n + b ) b ) + n ) α ) n...α q ) n β ) n...β s ) n ). n + Using 2.) and 3.2), we have, + n ) ) α ) n...α q ) n n + ka k β ) n...β s ) n

7 Generalized Dziok-Srivastava oerator 324 b ) + b ) + n ) ) α ) n...α q ) n n + β ) n...β s ) n ) b + b ) = b ) n + n + b. + n ) α ) n...α q ) n β ) n...β s ) n ) n + a k n + b ) b ) + n ) a)n c) n ) n + That is, ka k n + b ) b n + ) ) + n ) α ) n...α q ) n β ) n...β s ) n ) = δ, > b ). n + Thus, by the definition given by.5), f N n,δ h). This coletes the roof. Siilarly, we rove the following Theore. Theore 3.2. If 3.3) δ = μn + ) + ] b )n + )! then L, q,sα,β,n,,b; μ) N n,δ h). )] + + n ) α ) n...α q ) n β ) n...β s ) n ), μ >) n + 4. Further Neighborhood Proerties In this section, we deterine the neighborhood roerties for functions belonging to the subclasses H,,α q,s α,β,n,,b) and L,,α q,s α,β,n,,b; μ). For 0 α<and z U, a function f is said to be in the class Hq,s,,α α,β,n,,b) if there exists a function g Hq,s,α,β,n,,b) such that 4.) fz) gz) < α.

8 3242 H. Mahzoon For 0 α<and z U, a function f is said to be in the class L,,α q,s α,β,n,,b; μ) if there exists a function g L, q,sα,β,n,,b; μ) such that the inequality 4.) holds true. Theore 4.. If g H, q,s α,β,n,,b) and 4.2) α = δn + b ) n + ) n + b ) + n ) α ) n...α q ) n β ) n...β s ) n + n ) α ) n...α q ) n β ) n...β s ) n ) n + ) )], n + b then N n,δ g) Hq,s,,α α,β,n,,b). Proof. Let f N n,δ g). Then, 4.3) k a k b k δ, which yields the coefficient inequality, 4.4) a k b k δ n +, n N). Since g H, q,sα,β,n,,b)by3.2), we have, 4.5) b k n + b ) b ) + n ) α ) n...α q ) n β ) n...β s ) n ) n + so that,

9 fz) gz) < Generalized Dziok-Srivastava oerator 3243 a k b k b k δ n + n + b ) n + b ) + n ) α ) n...α q ) n β ) n...β s ) n + n ) α ) n...α q ) n β ) n...β s ) n ) n + ) )] n + b = α. Thus, by definition, f Hq,s,,α α,β,n,,b) for α given by 4.2). Thus the roof is colete. On siilar lines, we rove the following theore. Theore 4.2. If g L, q,s α,β,n,,b; μ) and 4.6) α = ζ where and ζ = δμn + ) + ] + n ) ) α ) n...α q ) n n + β ) n...β s ) n n + )Q Q = {μn + ) + } + n ) ) α ) n...α q ) n n + β ) n...β s ) n b ) +! ))] then N n,δ g) L,,α q,s α,β,n,,b; μ).

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