e Scientific World Journal Volume 2016, Article ID 6360250, 7 ages htt://dx.doi.org/10.1155/2016/6360250 Research Article On a New Class of -Valent Meromorhic Functions Defined in Conic Domains Mohammed Ali Alamri and Maslina Darus School of Mathematical Sciences,Faculty of Scienceand Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia Corresondence should be addressed to Maslina Darus; maslina@ukm.edu.my Received 26 Aril 2016; Acceted 27 June 2016 Academic Editor: Jin-Lin Liu Coyright 2016 M. A. Alamri and M. Darus. This is an oen access article distributed under the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original work is roerly cited. We define a new class of multivalent meromorhic functions using the generalised hyergeometric function. We derived this class related to conic domain. It is also shown that this new class of functions, under certain conditions, becomes a class of starlike functions. Some results on inclusion and closure roerties are also derived. 1. Introduction Let M denotetheclassoffunctionsoftheform f (z) = 1 z + a n z n, N =1,2,3,..., (1) which are analytic and -valent in the unctured unit disc centred at origin E = z : 0 < z < 1} = E \ 0}. Also by f(z) g(z) we mean f(z) is subordinate to g(z) which imlies the existence of an analytic function, called Schwartz function w(z) with w(z) < 1,forz E such that f(z) = g(w(z)), wheref(z) and g(z) are multivalent meromorhic functions. Note that if g is univalent in E then the above subordination is equivalent to f(0) = g(0) and f(e) g(e). The set of oints, for 0<γ<1and k [0,), where Ω k,γ =γω k +(1 γ), (2) Ω k =u+iv :u>k (u 1) 2 + V 2, u > 0}, (3) [1 showed that the extremal functions q k,γ (z) for conic regions are convex univalent and given by 1+(1 2γ)z, k=0, 1 z 1+ 2γ 1 k 2 [ 2 (arccos k) arctanh z, π 0<k<1, q k,γ (z) = 1+ 2γ 2 1+ z (log π2 1 z ), k=1, (4) 1+ γ k 2 1 [ sin ( π u(z)/ t 2R (t) 1 dx) 1, k>1, 0 1 x [ 2 1 (tx) 2 where R(t) is Legendre s comlete ellitic integral of the first kind with R (t) = 1 t 2 as its comlementary integral, u(z) = (z t)/(1 tz), t (0, 1), andz Eis chosen in such a way that k=cosh(πr (t)/r(t)).
2 The Scientific World Journal The generalised hyergeometric function q F s (α 1,...,α q ; β 1,...,β s ;z)for comlex arameters α 1,...,α q and β 1,...,β s with β j =0, 1, 2, 3,...,forj=1,2,3,...,s, is defined as q F (α 1 ) n (α q ) s (α 1,...,α q ;β 1,...,β s ;z)= n (β 1 ) n (β s ) n n! zn (α 1 ) n (α q ) =1+ n z n (β 1 ) n (β s ) n (n)!, with q s+1, q, s N 0 = N 0},and(α) n is the well-known Pochhammer symbol related to the factorial and the Gamma function by the relation (α) n = (α+n 1)! (α 1)! Also (5) imlies h (α 1,...,α q ;β 1,...,β s ;z) (5) = Γ (α+n). (6) Γ (α) =z q F s (α 1,...,α q ;β 1,...,β s ;z) M. Liu and Srivastava [2 defined a linear oerator for functions belonging to the class of multivalent meromorhic function H (α 1,...,α q ;β 1,...,β s ):M M as follows: H (α 1,...,α q ;β 1,...,β s ) =h (α 1,...,α q ;β 1,...,β s ;z) f(z). If we assume for brevity that H,q,s (α 1 ) = H (α 1,...,α q ; β 1,...,β s ) then the following identity holds for this oerator: z[h,q,s (α 1 )f(z) =α 1 H,q,s (α 1 +1)f(z) (α 1 +)H,q,s (α 1 )f(z). Shareef [3 defined and studied subclass MQ (k,λ,α 1 ) of meromorhic function associated with conic domain, for k 0, 0 λ<1,and 1, as follows: 1 [ z(h,q,s (α 1 )f(z)) +λz 2 (H,q,s (α 1 )f(z)) [(1 λ) H,q,s (α 1 )f(z) +λz(h,q,s (α 1 )f(z)) q k,γ (z). (7) (8) (9) (10) We now define a new subclass MQ (b,k,λ,α 1 ) of meromorhic function associated with conic domain, for k 0, 0 λ<1, 1,andb 1, as follows: 1 [ [1 + 1 b ( z(h,q,s (α 1 )f(z)) +λz 2 (H,q,s (α 1 )f(z)) (1 λ) H,q,s (α 1 )f(z) +λz(h,q,s (α 1 )f(z)) ) b q k,γ (z). (11) Since q k,γ is a convex and univalent function, for h(z) q k,γ (z) it means h(e ) is contained in q k,γ (E ),where h (z) = 1 [ [1 + 1 b ( z(h,q,s (α 1 )f(z)) +λz 2 (H,q,s (α 1 )f(z)) (1 λ) H,q,s (α 1 )f(z) +λz(h,q,s (α 1 )f(z)) ) b. (12) In the next two sections, for brevity, we dro the subscrits of the oerator H,q,s (α 1 ). 2. Preliminary Results Lemma 1 (see [4). Let h 2 (z) be convex in E and R(λh 2 (z) + μ) > 0,whereμ C, λ C\0},andz E.Ifh 1 (z) is analytic in E,withh 1 (0) = h 2 (0),then h 1 (z) + zh 1 (z) λh 1 (z) +μ h 2 (z) imlies h 1 (z) h 2 (z). (13) Lemma 2 (see [5). Let h(z) = 1 + c nz n and H(z) = 1+ d nz n and h H.IfH(z) is univalent and convex in E,then for n 1. c n d 1, (14) Lemma 3 (see [6). If q k,γ (z) = 1 + q 1 z+q 2 z 2 + then 8(1 γ)(arccos (k)) 2, 0<k<1, π 2 (1 k 2 ) 8(1 γ) q 1 = π 2, k=1, π 2 (1 γ) 4 t(k 2 1)k 2 (t)(1+t), k>1. One now states and roves the main results. 3. Main Results (15) In this section we exlore some of the geometric roerties exhibited by the class MQ (b,k,λ,α 1 ). We begin by discussing an inclusion roerty for the class MQ (b,k,λ,α 1 ). Theorem 4. If R(α 1 )>R(bq k,λ (z) 1) then MQ (b,k,λ,α 1 +1) MQ (b,k,λ,α 1 ). (16)
The Scientific World Journal 3 Proof. Let f MQ (b,k,λ,α 1 +1)and set 1 [1 + 1 b ( z(h(α 1)f(z)) +λz 2 (H (α 1 )f(z)) (1 λ) H(α 1 )f(z) +λz(h(α 1 )f(z)) ) b=h(z). (17) (α 1 +)(H(α 1 )f(z)), z[h(α 1 )f(z) +[H(α 1 )f(z) =α 1 H ((α 1 +1)f(z)) (α 1 +)(H(α 1 )f(z)), z 2 (H (α 1 )f(z)) =α 1 z(h(α 1 +1)f(z)) But differentiating (9) with resect to z we get [zh (α 1 )f(z) =α 1 H ((α 1 +1)f(z)) Putting(18)in(17)wehave (α 1 ++1)z(H(α 1 )f(z)). (18) λzα 1 (H (α+1) f (z)) +[1 λ(α 1 ++1)z(H(α 1 )f(z)) (1 λ) H(α 1 )f(z) +λz(h(α 1 )f(z)) = bh(z), λα 1 z(h(α 1 +1)f(z)) + (1 λ) α 1 (H (α 1 +1)f(z)) (1 λ) H(α 1 )f(z) +λz(h(α 1 )f(z)) = bh(z) +(α 1 +). (19) Taking logarithmic derivative of (19) we have z(h(α 1 +1)f(z)) +λz 2 (H (α 1 +1)f(z)) (1 λ) (H (α 1 +1)f(z)) + λz (H (α 1 +1)f(z)) bzh (z) +bh(z) +b(b 1) = bh (z) +α 1 +, z(h(α 1 +1)f(z)) +λz 2 (H (α 1 +1)f(z)) (1 λ) (H (α 1 +1)f(z)) + λz (H (α 1 +1)f(z)) = b[h(z) + 1 [1 zh (z) bh (z) +α 1 + b(b 1), + 1 b ( z(h(α 1 +1)f(z)) +λz 2 (H (α 1 +1)f(z)) (1 λ) H(α 1 +1)f(z) +λz(h(α 1 +1)f(z)) ) b=h(z) + zh (z) bh (z) +(α 1 +). Since f MQ q (b,k,λ,α 1 +1), therefore (20) rovided R( bq k,λ (z) + α 1 +) > 0or equivalently R(α 1 )> R(bq k,γ (z) 1).Hence,f MQ (b,k,λ,α 1 ). WenowshowthattheclassMQ (b,k,λ,α 1 ) is closed under a certain integral. Theorem 5. If f(z) MQ (b,k,λ,α 1 ), then the integral B η (H (α) f (z)) = η z η mas f(z) into MQ (b,k,λ,α 1 ). Proof. From (23) we have z t η 1 (H (α) f (z)) dt (23) 0 z z η B η (H (α) f (z)) = (η ) t η 1 (H (α) f (z)) dt. (24) 0 Note that B η (H (α) f (z)) = (H (α)) B η f (z). (25) h (z) + Using Lemma 2 we have zh (z) bh (z) +(α 1 +) q k,γ (z). (21) h (z) q k,γ (z), (22) Differentiating (24) above we get ηz η 1 B η (H (α) f (z))+z(b η (H (α) f (z))) =(η )z η 1 (H (α) f (z)),
4 The Scientific World Journal ηb η (H (α) f (z))+z(b η (H (α) f (z))) =(η )(H(α) f (z)). Differentiate again η(b η (H (α) f (z))) +z(b η (H (α) f (z))) +(B η (H (α) f (z))) =(η )(H(α) f (z)), z(b η (f (z))) =(η )(B η (H (α) f (z))) (η+1)(b η (f (z))). (26) (27) Now let 1 [ [1 + 1 b ( z(b η (H (α 1 )f)) +λz 2 (B η (H (α 1 )f)) (1 λ) B η (H (α 1 )f)+λz(b η (H (α 1 )f)) ) b =g(z). Using (26) and (27) in (28) we get (28) zb η (H (α 1 )f) +λ[z(η )(H(α 1 )f) z(1+η)(b η (H (α 1 )f)) (1 λ) B η (H (α 1 )f)+λz(b η (H (α 1 )f)) = bg(z) +b 2 b, λz(η )(H(α 1 )f) +[1 λ(1+η)[(η )(H(α 1 )f) η(b η (H (α 1 )f)) (1 λ) B η (H (α 1 )f)+λz(b η (H (α 1 )f)) = bg(z) +b 2 b, (η ) [λz (H (α 1 )f) (1 λ)(h (α 1 )f) ληz(b η (H (α 1 )f)) (1 λ) η(b η (H (α 1 )f)) (1 λ) B η (H (α 1 )f)+λz(b η (H (α 1 )f)) = bg(z) +b 2 b, (29) (η ) [λz (H (α 1 )f) (1 λ)(h (α 1 )f) (1 λ) B η (H (α 1 )f)+λz(b η (H (α 1 )f)) = bg(z) +b 2 b+η. Now taking logarithmic derivative we have (η ) [λz (H (α 1 )f) + λ (H (α 1 )f) + (1 λ) (H (α 1 )f) (η ) [λz (H (α 1 )f) (1 λ) (H (α 1 )f) (1 λ) B η (H (α 1 )f) +λz(b η (H (α 1 )f)) +λ(bη (H (α 1 )f)) bg (z) (1 λ) B η (H (α 1 )f)+λz(b η (H (α 1 )f)) = bg (z) +b 2 b, 1 [1 + 1 b ( z(h(α 1)f) +λz 2 (H (α 1 )f) zg (z) ) b=g(z) + (1 λ)(h (α 1 ) f) + λz (H (α 1 )f) bg (z) +b 2 b. (30) Using Lemma 2 we get g(z) q k,γ (z) which imlies This roves the assertion. B η (f (z)) MQ (b,k,λ,α 1 ). (31) Now we get coefficient estimates of the class MQ (b, k, λ, α 1 ). Theorem 6. If f(z) MQ (b,k,λ,α 1 ) and f(z) is given by (1) then a n bq 1 +b +n k, k=0 a 0 =1 (32) rovided 2(λ 1)+(3 n) λ<kλ for 1 λ + (n 1) λ>0, n 1
The Scientific World Journal 5 2(λ 1)+(3 n) λ>kλ for all k n. for 1 λ + (n 1) λ<0, (33) Proof. Let f(z) MQ (b,k,λ,α 1 ); then, by definition, we have 1 [1 + 1 b ( z(h(α 1)f(z)) +λz 2 (H (α 1 )f(z)) (1 λ) H(α 1 )f(z) +λz(h(α 1 )f(z)) b)=h(z) q k,γ (z), z(h(α 1)f(z)) +λz 2 (H (α 1 )f(z)) (1 λ) H(α 1 )f(z) +λz(h(α 1 )f(z)) which gives =bh(z) + b b2, z(h(α 1)f(z)) +λz 2 (H (α 1 )f(z)) =[bh(z) + [(1 λ) H(α 1 )f(z) +λz(h(α 1 )f(z)). (34) (35) Assuming h(z) = 1 + c nz n,then(35)becomes z (n ) a n z n +λ( (+1)z (n ) (n 1) + a n z n ), (1 λ λ) z n a n [1+λ(n 1)zn =[b + bc n z n + [(1 λ λ) z + a n [1+λ(n 1)z n. From (37) we have (1 λ λ) z n a n [1+λ(n 1)zn =[b+bc 1 z +bc 2 z 2 +bc 3 z 3 + +bc n z n + + [(1 λ λ)z (37) (38) Let us write H(α 1 )f(z) = f(z);then f (z) =z + a n z n, f (z) = z 1 + a n (n ) z n 1, zf (z) = z + a n (n ) z n, f (z) = ( 1)z 2 + a n (n ) (n 1) z n 2, (36) + a n [1+λ(n 1)z n. Now comaring coefficients of z 1 we have a 1 1 (1 λ) = a 1b(1 λ)+a 1 ( (1 λ)+bc 1 (1 λ λ), +b +1 a 1 (1 λ) [ =bc 1 (1 λ λ), ) (39) z 2 f (z) = ( 1)z + a n (n ) (n 1) z n. a 1 = (1 λ λ) (1 λ) [ +b +1 bc 1,
6 The Scientific World Journal andcomaringthecoefficientsofz 2 gives a 2 [ 2 [1+λ(1 )=a 2 [1+λ(1 )[b + +a 1 [1 λ bc 1 +bc 2 (1 λ λ), a 2 [1+λ(1 )[ 2 b + + =a 1 [1 λbc 1 +bc 2 (1 λ λ), (40) +b +2 a 2 [1+λ(1 )[ =a 1 [1 λbc 1 +bc 2 (1 λ λ), a 2 = (1 + λ (1 )) ( +b +2) ((1 λ λ)bc 2 +[1 λba 1 c 1 ), and for the coefficient of z 3 we have +b +3 a 3 [1+λ(2 )[ =bc 1 a 2 [1 + λ (1 ) + bc 2 a 1 [1 λ+bc 3 [1 λ λ, a 3 = (1 + λ (2 )) ( +b +3) ([1 λ λbc 3 +[1 λbc 2 a 1 +[1+λ(1 ) bc 1 a 2 ), which generalise to a n = (1 λ + (n 1) λ) ( +b +n) [(1 λ λ)bc n +(1 λ)bc n 1 a 1 +(1 λ+λ) bc n 2 a 2 + +(1 λ+(n 2) λ) bc 1 a n 1. The above exression can also be written as a n = (1 λ + (n 1) λ) ( +b +n) n k=1 Now taking [1 λ + (k 2) λ c n k 1 a k 1, with a 0 =1. (41) (42) (43) 1 λ+(k 2) λ A n,k =, n k, (44) 1 λ+(n 1) λ we have A n,k <1, if 2(λ 1)+(3 n) λ<kλ 2(λ 1)+(3 n) λ>kλ for 1 λ+(n 1) λ>0, for 1 λ+(n 1) λ<0, (45) for all n k.sinceq k,γ (z) is univalent and q k,γ (E) is convex, alying Rogosinski s theorem we have c n q 1, (46) where q 1 is given in (15). Under the conditions given in (45), exressions (39) (42) give a 1 bq 1 +b +1, a 2 bq 1 +b +2 (1 + a 1 ), a 3 bq 1 +b +3 (1 + a 1 + a 2 ),. a n bq 1 +b +n (1+ a 1 + a 2 + + a n 1 ). This can also be written as (47) a n bq 1 +b +n a k. (48) k=0 This concludes the roof. Cometing Interests The authors declare that there is no conflict of interests regarding the ublication of this aer. Acknowledgments n 1 The work here is suorted by AP-2013-009. References [1 S. Kanas and A. Wisniowska, Conic domains and k-starlike functions, Revue Roumaine de Mathematique Pures et Aliquees,vol.45,no.4,.647 657,2000. [2 J.-L. Liu and H. M. Srivastava, Classes of meromorhically multivalent functions associated with the generalized hyergeometric function, Mathematical and Comuter Modelling, vol. 39,no.1,.21 34,2004. [3 Z. Shareef, Somegeometricroertiesofcertainclassesofanalytic functions [Ph.D. thesis, Universiti Kebangsaan Malaysia, Bangi, Malaysia, 2015. [4 P. Eenigenburg, P. T. Mocanu, S. S. Miller, and M. O. Reade, On a briot-bouquet differential subordination, in General Inequalities,vol.3ofInternational Series of Numerical Mathematics,. 339 348, Birkhäuser, Basel, Switzerland, 1983.
The Scientific World Journal 7 [5 W. Rogosinski, On the coefficients of subordinate functions, Proceedings of the London Mathematical Society, Series 2,vol.48, no. 1,. 48 82, 1945. [6 F.M.Al-OboudiandK.A.Al-Amoudi, Onclassesofanalytic functions related to conic domains, Mathematical Analysis and Alications,vol.339,no.1,.655 667,2008.
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