Research Article Sufficient Conditions for λ-spirallike and λ-robertson Functions of Complex Order

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1 Mathematics Volume 2013, Article ID , 4 pages Research Article Sufficient Conditions for λ-spirallike and λ-robertson Functions of Complex Order B. A. Frasin Department of Mathematics, Faculty of Science, Al al-bayt University, P.O. Box , Mafraq, Jordan Correspondence should be addressed to B. A. Frasin; bafrasin@yahoo.com Received 7 January 2013; Revised 12 February 2013; Accepted 12 February 2013 Academic Editor: Abdellatif Agouzal Copyright 2013 B. A. Frasin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We obtain several sufficient conditions for λ-spirallike and λ-robertson functions of complex order by making use of the wellknown Jack Lemma. 1. Introduction and Definitions Let A denotetheclassoffunctionsoftheform f =z+ a n z n, (1) which are analytic in the open unit disk U ={z:z Cand z < 1}. Afunctionf A is said to be the λ-spirallike function of complex order b and type α(0 α<1)in U, denoted by S λ α (b) if and only if Re {1+ n=2 (zf 1)} >α (z U), (2) f for some real numbers λ R with λ < π/2 and b C := C \{0}. Furthermore, a function f A is also said to be the λ-robertson function of complex order b and type α(0 α<1)in U if and only if Re{1+ (zf f )} > α (z U), (3) for some real numbers λ R with λ < π/2 and b C.We denote this class by C λ α (b). Noting that the above function classes include several subclasses which have important role in the analytic and geometric function theory. From this reason, we want to state some of them. (i) S λ 0 (b) =: Sλ (b) (b C ; λ < π/2) is the class of the λ-spirallike function of complex order b introduced and studied by Al-Oboudi and Haidan [1]. (ii) C λ 0 (b) =: Cλ (b) (b C ; λ < π/2) is the class of the λ-robertson function of complex order b (see, [2]). (iii) S 0 α (b) =: S α (b) (b C ; 0 α < 1) is the class of starlike functions of complex order b and type α introduced and studied by the author [3], and S α (1) =: S α is said to be the starlike functions class of order α(0 α<1)andwas studied by Robertson [4]. (iv) C 0 α (b) =: C α(b) (b C ; 0 α < 1) is the class of convex functions of complex order b and type α introduced and studied by the author [3], and C α (1) =: C α is said to be the convex functions class of order α(0 α<1)andwas studied by Robertson [4]. (v) S λ 0 (1) =: Sλ ( λ < π/2) is known the λ-spirallike univalent functions class and was defined by Spacek [5], S 0 0 (b) =: S(b) (b C ) is said to be the starlike functions class of complex order and was studied by Nasr and Aouf [6], and S λ 0 (1 β) =: Sλ (β) (0 β < 1) is known the λ-spirallike functions class of order β and was studied by Libera [7]. (vi) C λ 0 (1) =: Cλ ( λ < π/2) is known the λ-robertson type functions class and was first studied by Robertson [8], C 0 0 (b) =: C(b) is called the convex

2 2 Mathematics functions class of complex order and was studied by Wiatrowski [9], Nasr and Aouf [10] and Aouf[11], and C λ 0 (1 β) =: Cλ (β) (0 β < 1) is known the λ-robertson type functions class of order β and was studied by Chichra [12]. In this paper, we obtain several sufficient conditions for the analytic functions f belonging to the classes S λ α (b), Cλ α (b), S α (b), C α(b), S(b), C(b), S α, and C α by making use of the well-known Jack Lemma [13]. 2. Main Result In order to derive our main result, we have to recall here the following Jack Lemma. Lemma 1 (see [13]). Let w be analytic in U such that w(0) = 0. Then,if w attains its maximum value on circle z = r < 1 at a point z 0 U,onehas where k>1is a real number. z 0 w (z 0 ) =kw(z 0 ), (4) Now, with the help of Lemma 1, we can prove the following result. Theorem 2. Let f A,, b C and λ < π/2, andlet F be defined by F = (1 μ) f +μzf, (0 μ 1). (5) If F satisfies any of the following inequalities: (1 + 1 (zf 1)) F ( z[zf ] ( zf 2 F F ) ) b < (1 α) cos λ, (z U; 0 α<1), Re {((1+(e iλ /)((zf /F ) 1)) 1 (z[zf ] /F ) (zf /F ) 2 ) ((1+(e iλ /)((zf /F ) 1)) 1) 1 } cos λ < 2 Re {be iλ }, (z U), (10) then Re (1+ (zf F 1)) >α, (z U; 0 α<1). The powers are taken by their principal value. (9) (11) Proof. Define a function w by (1 + (zf 1)) =1+(1 α) w. (12) F Then, w is analytic in U and w(0) = 0.Itfollowsfrom (12)that ((z[zf ] /F ) (zf /F ) 2 ) (1 + (e iλ /) ((zf /F ) 1)) ((1 + (e iλ /) (zf /F 1)) 1) < ( ), (z U; 0 α<1), 2 α ((z[zf ] /F ) (zf /F ) 2 ) 1+(e iλ /) ((zf /F ) 1) < ((1 α) ), (z U; 0 α<1), 2 α ((z[zf ] /F ) (zf /F ) 2 ) (6) (7) ((z[zf ] /F ) (zf /F ) 2 ) 1+(e iλ /) ((zf /F ) 1) +(1+ (zf 1)) 1 F = (1 α) w [1 + be iλ cos λ Thus, we have zw w 1 1+(1 α) w ]. (13) ((z[zf ] /F ) (zf /F ) 2 ) (1 + (e iλ /) ((zf /F ) 1)) F 1 = ((1 + (e iλ /) ((zf /F ) 1)) 1) ((1 + (e iλ /) ((zf /F ) 1)) +1 ) < ((1 α) (2 α) 2 ), (z U; 0 α<1), (8) zw w 1 1+(1 α) w, (14)

3 Mathematics 3 (z[zf ] /F ) (zf /F ) 2 F 2 = 1+(e iλ /) ((zf /F ) 1) ( (1 α) zw 1+(1 α) w ), (z[zf ] /F ) (zf /F ) 2 F 3 = (1 + (e iλ /) ((zf /F ) 1)) +1 F 4 =(1+ ( (1 α) zw [1+(1 α) w ] 2 ), 1 (zf 1)) F ( z[zf ] F ( zf 2 F ) ) (1 α) zw, F 5 =(1+(e iλ /)((zf /F ) 1)) 1 ((z[zf ] /F ) (zf /F ) 2 ) (15) (16) (17) ((1+(e iλ /)((zf /F ) 1)) 1) 1 zw w. (18) F 3 (z 0 ) = be iλ cos λ ( (1 α) z 0w (z 0 ) [1 + (1 α) w(z 0 )] 2 ) b k (1 α) = (cos λ) w(z 0) [ 1+(1 α) w(z 0) ]2 ((1 α) (2 α) 2 ), F 4 (z 0 ) = be iλ cos λ (1 α) z 0 w (z 0 ) b = cos λ k (1 α) w(z 0) b (1 α) cos λ, Re {F 5 (z 0 )} = Re { be iλ cos λ z 0 w (z 0 ) w(z 0 ) } =k(cos λ) Re { e iλ } 2 Re {be iλ }, (19) which contradicts our assumptions (6) (10), respectively. Therefore, w < 1 holds true for all z U. Wefinally have (1+ (zf 1)) 1 <1 α, (20) F thus, we have Re (1+ (zf F 1)) >α, (z U; 0 α<1). (21) We claim that w < 1 in U. Forotherwise,by Lemma 1, there exists z 0 U such that z 0 w (z 0 )=kw(z 0 ), where w(z 0 ) = 1 and k>1. Therefore, (14) (18)yield F 1 (z 0 ) = be iλ cos λ z 0 w (z 0 ) 1 w(z 0 ) 1+(1 α) w(z 0 ) b k = (cos λ) w(z 0) 1+(1 α) w(z 0) ( 2 α ), F 2 (z 0 ) = be iλ cos λ ( (1 α) z 0w (z 0 ) 1+(1 α) w(z 0 ) ) b k (1 α) = (cos λ) w(z 0) 1+(1 α) w(z 0) ((1 α) ), 2 α Remark 3. Taking different choices of α, λ, μ, and in Theorem 2, we obtain new sufficient conditions for functions to be in the classes S λ α (b), Cλ α (b), S α (b), C α(b), S(b), C(b), S α,andc α. Acknowledgment The author would like to thank the referee for his helpful comments and suggestions. References [1] F. M. Al-Oboudi and M. M. Haidan, Spirallike functions of complex order, Natural Geometry, vol. 19, no. 1-2, pp , [2] M. K. Aouf, F. M. Al-Oboudi, and M. M. Haidan, On some results for λ-roberston functions ofcomplex order, Publications de l Institut Mathématique,vol.75,no.91,pp.93 98,2005. [3] B. A. Frasin, Family of analytic functions of complex order, Acta Mathematica,vol.22,no.2,pp ,2006. [4] M. I. S. Robertson, On the theory of univalent functions, Annals of Mathematics,vol.37,no.2,pp ,1936.

4 4 Mathematics [5] L. Spacek, Contribution a la theori des function univalentes, Casopsis pro Pestovani Matematiky a Fysiki, vol.62,pp.12 19, [6] M.A.NasrandM.K.Aouf, Starlikefunctionofcomplexorder, The Natural Sciences and Mathematics, vol.25,no.1, pp. 1 12, [7] R. J. Libera, Univalentα-spiral functions, Canadian Mathematics,vol.19,pp ,1967. [8] M. S. Robertson, Univalent functions f for which zf is spirallike, The Michigan Mathematical Journal, vol.16,pp , [9] P. Wiatrowski, The coefficients of a certain family of holomorphic functions, Zeszyty Naukowe Uniwersytetu Lodzkiego, Nauki Matematyczno Przyrodnicze, Seria 2, Zeszyt, no. 39, pp , [10] M. A. Nasr and M. K. Aouf, On convex functions of complex order, Mansoura Science Bulletin, vol. 9, pp , [11] M. K. Aouf, p-valent classes related to convex functions of complex order, The Rocky Mountain Mathematics, vol.15,no.4,pp ,1985. [12] P. N. Chichra, Regular functions f for which zf is αspiral-like, Proceedings of the American Mathematical Society, vol. 49, pp , [13] I. S. Jack, Functions starlike and convex of order α, the London Mathematical Society,vol.3,pp ,1971.

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