Sufficient conditions for certain subclasses of meromorphic p-valent functions

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1 Bol. Soc. Paran. Mat. (3s.) v. 33 (015): c SPM ISSN on line ISSN in ress SPM: doi:10.569/bsm.v33i.1919 Sufficient conditions for certain subclasses of meromorhic -valent functions Onkar Singh, Pranay Goswami and Basem Frasin abstract: In the resent aer, we obtain certain sufficient conditions for meromorhic -valent functions. Several corollaries and consequences of the main results are also considered. Key Words: Meromorhic multivalent functions, meromorhic starlike functions, meromorhic convex functions, meromorhic close-to-convex functions. Contents 1 Introduction and definitions 9 Main Results 10 3 Corollaries and Consequences Introduction and definitions Let Σ denote the class of functions of the form f = 1 z + a n z n, ( N := {1,,3,...}), (1.1) n= which are analytic and -valent in the unctured oen unit disk U = {z : z C;0 < z < 1} =: U\{0}. where U is an oen unit disk. A function f in Σ is said to be meromorhically -valent starlike of order δ if and only if { } R zf > δ (z U ), (1.) f for some δ( 0 δ < ). We denote by Σ (δ) the class of all meromorhically -valent starlike of order δ. Further, a function f in Σ is said to be meromorhically -valent convex of order δ if and only if { } R 1 zf f > δ (z U ), (1.3) 000 Mathematics Subject Classification: 30C45 9 Tyeset by B S P M style. c Soc. Paran. de Mat.

2 10 Onkar Singh, Pranay Goswami and Basem Frasin for some δ( 0 δ < ). We denote by Σ k (δ) the class of all meromorhically -valent convex of order δ. A function f belonging to Σ is said to be meromorhically -valent close-to-convex of order δ if it satisfies R f z 1 > δ (z U ), (1.4) for some δ(0 δ < ).We denote byσ c (δ) the subclass ofσ consisting of functions which are meromorhically -valent close-to-convex of order δ in U. Note that Σ 1 (δ) = Σ (δ), Σ k 1 (δ) = Σk (δ) and Σ c 1 (δ) = Σc (δ), where Σ (δ), Σ k (δ) and Σ c (δ) are subclasses of Σ 1 consisting meromorhic univalent functions which are resectively, starlike, convex and close-to-convex of order δ(0 δ < 1). Some subclasses of Σ = Σ when = 1 were considered by (for examle) Miller [1], Pommerenke [16], Clunie [7], Frasin and Darus [8] and Royster [17]. Furthermore, several subclasses of Σ were studied by (amongst others) Mogra et al. [14], Goyal and Prajaat [11], Owa et al. [15], Srivastava et al. [18], Wang and Zhang [1],Uralegaddi and Ganigi [19], Cho et al. [6], Aouf [1-4], and Uralegaddi Somantha [0]. The object in the resent aer is to obtain some sufficient conditions for meromorhic -valent functions. In the roofs of our main results, we need the following Jack s Lemma [9]: Lemma 1.1. Let the (non constant) function w be analytic in U with w(0) = 0. If w attains its maximum value on the circle z = r < 1 at a oint z 0 U, z 0 w (z 0 ) = mw(z 0 ) where m is a real number and m n where n 1.. Main Results With the aid of Lemma 1.1, we derive the next two theorems. Theorem.1. Let the function f Σ, satisfies the inequality [ ] R α zf f +β 1+ zf f > [(α+β)+n]+λ[(α+β) n]. (.1) (1+λ) R [ (z f) α ( z +1 f ) β ] > 1+λ (.) where (α,β R, λ 1,,n N).

3 Sufficient conditions for certain subclasses of meromorhic Proof: Let the function w be defined by ( (z f) α z +1 f ) β = 1+λw 1+w. (.3), clearly, w is analytic in U with w(0) = 0. We also find from (.3) that [ ] α zf f +β 1+ zf f = (α+β) λzw 1+λw + zw,(z U). (.4) 1+w Suose there exists a oint z 0 U such that w(z 0 ) = 1 and w < 1, when z < z 0. by alying Lemma 1.1, there exists m n such that z 0 w (z 0 ) = mw(z 0 ), ( m n 1;w(z0 ) = e iθ ;θ R ). (.5) by using (.4) and (.5), it follows that [ ] R α zf (z 0 ) f(z 0 ) +β 1+ zf (z 0 ) f (z 0 ) λme iθ me iθ = (α+β) R 1+λe iθ +R 1+e iθ = (α+β) λm(λ+cosθ) 1+λ +λcosθ + m m(λ 1) = (α+β) (1+λ +λcosθ) (α+β) n λ 1 1+λ [(α+β)+n]+λ[(α+β) n] (1+λ) which contradicts the given hyothesis. Hence w < 1, which imlies 1 (z f) α z +1 f β (z f) α β < 1 (.6) z +1 f λ or equivalently R [ (z f) α ( z +1 f ) β ] > 1+λ. This comletes the roof of Theorem.1.

4 1 Onkar Singh, Pranay Goswami and Basem Frasin Theorem.. Let the function f Σ, satisfies the inequality [ ] R α zf f +β 1+ zf f < {(α+β)+n}λ+{(α+β)+n}. (3.1) λ+ R [ (z f) α ( z +1 ) β ] f > 1 +λ (3.) where (α,β R, λ 1,,n N). Proof: Let the function w be defined by ( (z f) α z +1 β f ) = clearly w is analytic in U with w(0) = 0 Using logarithmic differentiation (3.3) yields 1 (1+λ)w+1. (3.3) [ ] α zf f +β 1+ zf f = (α+β)+ (1+λ)zw,(z U). (3.4) 1+(1+λ)w Suose there exists a oint z 0 U such that w(z 0 ) = 1 and w < 1, when z < z 0. by alying Lemma 1.1, there exists m n such that z 0 w (z 0 ) = mw(z 0 ), ( m n 1;w(z0 ) = e iθ ;θ R ). (3.5) by using (3.4) and (3.5), it follows that [ ] R α zf (z 0 ) f(z 0 ) +β 1+ zf (z 0 ) f (z 0 ) ( (1+λ)z0 w ) (z 0 ) = (α+β)+r (1+λ)w(z 0 )+1 (1+λ)me iθ = (α+β)+r (1+λ)e iθ +1 m(1+λ)(1+λ+cosθ) 1+(1+λ) +(1+λ)cosθ {(α+β)+n}λ+{(α+β)+n} λ+ = (α+β)+ which contradicts the hyothesis (3.1). It follows that w < 1, that is 1 (z f) α β 1 < 1+λ. z +1 f This evidently comletes the roof of Theorem..

5 Sufficient conditions for certain subclasses of meromorhic Corollaries and Consequences In this concluding section, we consider some Corollaries and Consequences of our main results (Theorem.1 and Theorem.). Uon setting α = 0 and β = 1 in Theorem.1, we get Corollary 3.1. If the function f Σ satisfies the inequality R 1+ zf f > (+n)+λ( n) (λ 1,,n N) (1+λ) ( z +1 f ) R > 1+λ. Setting = n = 1 in Corollary 3.1, the result reduces to Corollary 3.. If the function f Σ satisfies the inequality R 1+ zf f > 3+λ (λ 1) (1+λ) or equivalently, R [ z f ] > 1+λ, 1+λ f Σ c. Setting α = 0 and β = 1, Theorem.1 gives Corollary 3.3. Let the function f Σ, satisfies the inequality ( zf ) R > (+n)+λ( n) (λ 1,,n N). f (1+λ) R(z f) > 1+λ. Setting = n = 1 in Corollary 3.3, the result reduces to Corollary 3.4. Let the function f Σ, satisfies the inequality ( zf ) R > 3+λ (λ 1). f (1+λ) R(zf) > 1+λ.

6 14 Onkar Singh, Pranay Goswami and Basem Frasin Setting α = 1 γ and β = γ; γ R in Theorem., we obtain the following secial case: Corollary 3.5. Let the function f Σ, satisfies the inequality [ ] R (1 γ) zf f +γ 1+ zf f > + n 1 λ (λ 1,,n N). 1+λ [ R (z f) ( zf ) γ ] f Setting α = 0 and β = 1 in Theorem., we get > 1+λ. Corollary 3.6. If the function f Σ satisfies the inequality [ ] R 1+ zf f < (+n)λ+(+n) (λ 1,,n N) λ+ [] z +1 R f > 1 +λ. Setting = n = 1 in Corollary 3.6, the result reduces to Corollary 3.7. If the function f Σ satisfies the inequality R 1+ zf f < λ+3 (λ 1) λ+ or equivalently, R [( z f )] > 1 +λ, 1 f Σ c. +λ Setting α = 0 and β = 1, Theorem., it gives Corollary 3.8. Let the function f Σ, satisfies the inequality ( zf ) R < (+n)λ+(+n) (λ 1,,n N). f λ+ R[(z f)] > 1 +λ.

7 Sufficient conditions for certain subclasses of meromorhic Setting = n = 1 in Corollary 3.8, the result reduces to Corollary 3.9. Let the function f Σ, satisfies the inequality ( zf ) R < 3+λ (λ 1). f +λ R[(zf)] > 1 +λ. Acknowledgments The authors would like to thank the referee for his helful comments and suggestions. References 1. M. K. Aouf, A generalization of meromorhic multivalent functions with ositive coefficients, Math. Jaon., 35 (1990), M. K. Aouf, On a class of meromorhic multivalent functions with ositive coefficients, Math. Jaon., 35 (1990), M.K. Aouf, On a certain class of meromorhic univalent functions with ositive coefficients, Rend. Mat. Al. (7) 11 (1991), M. K. Aouf, Certain classes of meromorhic multivalent functions with ositive coefficients, Math. Comut. Modelling 47(008), S.K. Bajai, A note on a class of meromorhic univalent functions, Rev. Roumaine Math. Pures Al. (1977), N.E. Cho, S.H. Lee and S. Owa, A class of meromorhic univalent functions with ositive coefficients, Kobe J. Math. 4 (1987), J. Clunie, On meromorhic schlicht functions, J. London Math. Soc. 34 (1959), B.A. Frasin and M. Darus, On certain meromorhic functions with ositive coefficients, Southeast Asian Bulletin of Mathematics 30 (006), I. S. Jack, Functions starlike and convex of order α, J. London Math. Soc. ()3, (1971), R.M. Goel and N.S. Sohi, On a class of meromorhic functions, Glas. Mat. Ser.III 17(37) (198), S. P. Goyal and J. K. Prajaat, A new class of meromorhic multivalent functions Involving certain linear oerator, Tamsui Oxford Journal of Mathematical Sciences 5() (009) J. Miller, Convex meromorhic maings and related functions, Proc. Amer. Math. Soc. 5 (1970), S.S. Miller and P.T. Mocanu, Differential subordinations and inequalities in the comlex lane, J. Differ. Equations, 67(1987), M.L. Mogra, T.R. Reddy and O.P. Juneja, Meromorhic univalent functions with ositive coefficients, Bull. Austral. Math. Soc. 3 (1985), S. Owa, H. E. Darwish and M. K. Aouf, Meromorhic multivalent functions with ositive and fixed second coefficients, Math. Jaon., 46() (1997), CH. Pommerenke, On meromorhic starlike functions, Pacific J. Math.13 (1963), 1-35.

8 16 Onkar Singh, Pranay Goswami and Basem Frasin 17. W.C. Royster, Meromorhic starlike multivalent functions, Trans. Amer. Math.Soc.107 (1963), H.M. Srivastava, H.M. Hossen and M.K. Aouf, A unified resentation of some classes of meromorhically multivalent functions, Comuter and Mathematics with Alications 38(1999), B.A. Uralegaddi and M.D. Ganigi, A certain class of meromorhic univalent functions with ositive coefficients, Pure Al. Math. Sci. 6 (1987), B.A. Uralegaddi and C. Somantha, New criteria for meromorhic starlike univalent functions, Bull. Austral. Math. Soc. 43 (1991), Z. Wang, Y. Sun and Z. Zhang, Certain classes of meromorhic multivalent functions, Comuters and Mathematics with Alications 58 (009) Onkar Singh Deartment of Mathematics, University of Rajasthan, Jaiur address: onkarbhati@gmail.com and P. Goswami Deartment of Mathematics, School of Liberal Studies, Bharat Ratn Dr. B.R. Amedkar University, Delhi , India address: ranaygoswami83@gmail.com and Basem Frasin Faculty of Science, Deartment of Mathematics, Al al-bayt University, P.O. Box: Mafraq, Jordan address: bafrasin@yahoo.com