Certain subclasses of uniformly convex functions and corresponding class of starlike functions

Transcription

1 Malaya Journal of Matematik 1(1)(2013) Certain subclasses of uniformly convex functions and corresponding class of starlike functions N Magesh, a, and V Prameela b a PG and Research Department of Mathematics, Govt Arts College for Men, Krishnagiri , India b Department of Mathematics, Adhiyamaan College of Engineering (Autonomous), Hosur , Tamilnadu, India Abstract In this paper, we defined a new subclass of uniformly convex functions and corresponding subclass of starlike functions with negative coefficients and obtain coefficient estimates Further we investigate extreme points, growth and distortion bounds, radii of starlikeness and convexity and modified Hadamard products Keywords: Univalent functions, convex functions, starlike functions, uniformly convex functions, uniformly starlike functions 2010 MSC: 30C45 c 2012 MJM All rights reserved 1 Introduction Denoted by S the class of functions of the form f(z) z + a n z n (11) that are analytic and univalent in the unit disc U {z : z < 1} and by ST and CV the subclasses of S that are respectively, starlike and convex Goodman [5, 6] introduced and defined the following subclasses of CV and ST A function f(z) is uniformly convex (uniformly starlike) in U if f(z) is in CV (ST ) and has the property that for every circular arc γ contained in U, with center ξ also in U, the arc f(γ) is convex (starlike) with respect to f(ξ) The class of uniformly convex functions denoted by UCV and the class of uniformly starlike functions by UST (for details see [5]) It is well known from [8, 11] that } zf (z) f UCV {1 f (z) Re zf (z) + f (z) In [11], Running introduced a new class of starlike functions related to UCV and defined as { } zf (z) f S p f(z) 1 Re zf (z) f(z) Note that f(z) UCV zf (z) S p Further Running generalized the class S p by introducing a parameter α, 1 α < 1, { } zf (z) f S p (α) f(z) 1 Re zf (z) f(z) α Corresponding author addresses: nmagi 2000@yahoocoin (N Magesh) and vasuprameelak@gmailcom (V Prameela)

2 N Magesh et al / Uniformly convex functions 19 Motivated by the works of Bharati et al [2], Frasin [3, 4], Murugusundaramoorthy and Magesh [10] and others [5, 6, 8, 11, 12, 17, 18], we define the following class: For β 0, 1 α < 1 and 0 λ < 1, we let S(λ, α, β, j) denote the subclass of S consisting of functions f(z) of the form (11) and satisfying the analytic criterion { zf (z) + (2λ 2 λ)z 2 f } (z) Re 4(λ λ 2 )z + (2λ 2 λ)zf (z) + (2λ 2 3λ + 1)f(z) α (12) > β zf (z) + (2λ 2 λ)z 2 f (z) 4(λ λ 2 )z + (2λ 2 λ)zf (z) + (2λ 2 3λ + 1)f(z) 1, z U (13) We also let T S(λ, α, β, j) S(λ, α, β, j) T where T, the subclass of S consisting of functions of the form f(z) z a n z n, a n 0, n j + 1 (14) introduced and studied by Silverman [14] We note that, by specializing the parameters j, λ, α, and β we obtain the following subclasses studied by various authors 1 T S(0, α, 0, 1) T (α) and T S(1, α, 0, 1) K(α) (Silverman [14]) 2 T S(0, α, 0, j) T (α, j) and T S(1, α, 0, j) K(α, j) (Srivastava et al [15]) 3 T S(1/2, α, 0, 1) P(α) (Al-Amiri [1], Gupta and Jain [7] and Sarangi and Uralegaddi [13]) 4 T S(λ, α, 0, j) B T (λ, α, j) (Frasin [3, 4] and Magesh [9]) 5 T S(1/2, α, β, 1) T R(α, β) (Rosy [12] and Stephen and Subramanian [16]) 6 T S(0, α, β, 1) T S(α, β) and T S(1, α, β, 1) UCV (α, β) (Bharati et al [2]) 7 T S(0, 0, β, 1) T S p (β) ( Subramanian et al [17]) 8 T S(1, 0, β, 1) UCV (β) (Subramanian et al [18]) The main object of this paper is to obtain a necessary and sufficient conditions for the functions f(z) in the generalized class T S(λ, α, β, j) Further we investigate extreme points, growth and distortion bounds, radii of starlikeness and convexity and modified Hadamard products for class T S(λ, α, β, j) 2 Coefficient Estimates In this section we obtain a necessary and sufficient condition for functions f(z) in the classes T S(λ, α, β, j) Theorem 21 A function f(z) of the form (11) is in S(λ, α, β, j) if where a n, (21) M n (2λ 2 λ)n 2 + (1 + λ 2λ 2 )n, F n (2λ 2 λ)n + (1 + 2λ 2 3λ) (22) and 1 α < 1, 1 2 λ < 1, β 0 Proof It suffices to show that β zf (z) + (2λ 2 λ)z 2 f (z) 4(λ λ 2 )z + (2λ 2 λ)zf (z) + (2λ 2 3λ + 1)f(z) 1 { zf (z) + (2λ 2 λ)z 2 f } (z) Re 4(λ λ 2 )z + (2λ 2 λ)zf (z) + (2λ 2 3λ + 1)f(z) 1

3 20 N Magesh et al / Uniformly convex functions We have β zf (z) + (2λ 2 λ)z 2 f (z) 4(λ λ 2 )z + (2λ 2 λ)zf (z) + (2λ 2 3λ + 1)f(z) 1 { zf (z) + (2λ 2 λ)z 2 f } (z) Re 4(λ λ 2 )z + (2λ 2 λ)zf (z) + (2λ 2 3λ + 1)f(z) 1 (1 + β) zf (z) + (2λ 2 λ)z 2 f (z) 4(λ λ 2 )z + (2λ 2 λ)zf (z) + (2λ 2 3λ + 1)f(z) 1 (1 + β) (M n F n ) a n a n This last expression is bounded above by () if and hence the proof is complete a n, Theorem 22 A necessary and sufficient condition for f(z) of the form (14) to be in the class T S(λ, α, β, j), is that a n (23) Proof In view of Theorem 21, we need only to prove the necessity If f(z) T S(λ, α, β, j) and z is real then M n a n z n 1 α β F n a n z n 1 (M n F n ) a n z n 1 Letting z 1 along the real axis, we obtain the desired inequality Finally, the function f(z) given by F n a n z n 1 a n, 1 α < 1, β 0 f(z) z where M j+1 and F j+1 as written in (22), is extremal for the function [M j+1 (1 + β) (α + β)f j+1 ] zj+1, (24) Corollary 23 Let the function f(z) defined by (14) be in the class T S(λ, α, β, j) Then a n This equality in (25) is attained for the function f(z) given by (24), n j + 1 (25) 3 Growth and Distortion Theorem Theorem 31 Let the function f(z) defined by (14) be in the class T S(λ, α, β, j) Then for z < r 1 r [M j+1 (1 + β) (α + β)f j+1 ] rj+1 f(z) r + [M j+1 (1 + β) (α + β)f j+1 ] rj+1 (31) The result (31) is attained for the function f(z) given by (24) for z ±r

4 N Magesh et al / Uniformly convex functions 21 Proof Note that [M j+1 (1 + β) (α + β)f j+1 ] a n this last inequality follows from Theorem 22 Thus Similarly, f(z) z f(z) z + This completes the proof a n z n r r j+1 a n z n r + r j+1 a n r a n r + a n, [M j+1 (1 + β) (α + β)f j+1 ] rj+1 [M j+1 (1 + β) (α + β)f j+1 ] rj+1 Theorem 32 Let the function f(z) defined by (14) be in the class T S(λ, α, β, j) Then for z < r 1 r (j + 1)() [M j+1 (1 + β) (α + β)f j+1 ] rj f (j + 1)() (z) r + [M j+1 (1 + β) (α + β)f j+1 ] rj (32) Proof We have and In view of Theorem 22, f (z) 1 na n z n 1 1 r j f (z) 1 + na n z n r j na n (33) na n (34) or, equivalently [M j+1 (1 + β) (α + β)f j+1 ] j + 1 na n na n a n, (35) (j + 1)() [M j+1 (1 + β) (α + β)f j+1 ] (36) A substitution of (36) into (33) and (34) yields the inequality (32) This completes the proof Theorem 33 Let f j (z) z, and f n (z) z zn, n j + 1 (37) for 0 λ 1, β 0, 1 α < 1 Then f(z) is in the class T S(λ, α, β, j) if and only if it can be expressed in the form f(z) µ n f n (z), (38) where µ n 0(n j) and nj µ n 1 Proof Assume that f(z) µ j f j (z) + µ n z nj nj µ n [z ] zn µ nz n

5 22 N Magesh et al / Uniformly convex functions Then it follows that µ n µ n 1, so by Theorem 22, f(z) T S(λ, α, β, j) Conversely, assume that the function f(z) defined by (14) belongs to the class T S(λ, α, β, j), then Setting µ n [Mn(1+β) (α+β)fn] 1 α Then (38) gives and hence the proof is complete a n, n j + 1 a n, (n j + 1) and µ j 1 µ n, we have, f(z) z f(z) z f(z) z + a n z n µ nz n (39) f j (z)µ j + µ n f n (z) nj (f n (z) z)µ n f n (z)µ n 4 Radii of close-to-convexity, Starlikeness and Convexity In this subsection, we obtain the radii of close-to-convexity, starlikeness and convexity for the class T S(λ, α, β, j) Theorem 41 Let f T S(λ, α, β, j) Then f(z) is close-to-convex of order σ (0 σ < 1) in the disc z < r 1, where [ ] 1 (1 σ)[mn n 1 (1 + β) (α + β)f n ] r 1 : inf, n j + 1 (41) n() The result is sharp, with extremal function f(z) given by (24) Proof Given f T, and f is close-to-convex of order σ, we have For the left hand side of (42) we have The last expression is less than 1 σ if f (z) 1 < 1 σ (42) f (z) 1 Using the fact, that f T S(λ, α, β, j), if and only if na n z n 1 n 1 σ a n z n 1 < 1 a n 1 ()

6 N Magesh et al / Uniformly convex functions 23 We can say (42) is true if Or, equivalently, which completes the proof n 1 σ z n 1 [M n(1 + β) (α + β)f n ] () [ ] (1 z n 1 σ)[mn (1 + β) (α + β)f n ], n() Theorem 42 Let f T S(λ, α, β, j) Then (i) f is starlike of order σ(0 σ < 1) in the disc z < r 2 ; where r 2 inf [( 1 σ n σ ) [Mn (1 + β) (α + β)f n ] () ] 1 n 1, n j + 1, (43) (ii) f is convex of order σ (0 σ < 1) in the unit disc z < r 3, where [( 1 σ r 3 inf n(n σ) ) [Mn (1 + β) (α + β)f n ] () ] 1 n 1, n j + 1 (44) Each of these results are sharp for the extremal function f(z) given by (24) Proof (i) Given f T, and f is starlike of order σ, we have For the left hand side of (45) we have The last expression is less than 1 σ if zf (z) f(z) 1 zf (z) f(z) 1 < 1 σ (45) Using the fact, that f T S(λ, α, β, j) if and only if We can say (45) is true if Or, equivalently, (n 1)a n z n 1 a n z n 1 n σ 1 σ a n z n 1 < 1 a n 1 () n σ 1 σ z n 1 < [M n(1 + β) (α + β)f n ] () [( ) ] 1 σ z n 1 [Mn (1 + β) (α + β)f n ] n σ () which yields the starlikeness of the family (ii) Using the fact that f is convex if and only if zf is starlike, we can prove (ii), on lines similar to the proof of (i)

7 24 N Magesh et al / Uniformly convex functions 5 Modified Hadamard Product Let the functions f i (z)(i 1, 2) be defined by f i (z) z a n,i z n, a n,i 0; j N, (51) then we define the modified Hadamard product of f 1 (z) and f 2 (z) by (f 1 f 2 )(z) z a n,1 a n,2 z n (52) Now, we prove the following Theorem 51 Let each of the functions f i (z)(i 1, 2) defined by (51) be in the class T S(λ, α, β, j) Then (f 1 f 2 ) T S(λ, δ 1, β, j), for The result is sharp δ 1 [M n(1 + β) (α + β)f n ] 2 [M n (1 + β) βf n ]() 2 2 F n () 2 (53) Proof We need to prove the largest δ 1 such that [M n (1 + β) (δ 1 + β)f n ] a n,1 a n,2 1 (54) 1 δ 1 From Theorem 22, we have and a n,1 1 a n,2 1, by the Cauchy-Schwarz inequality, we have an,1 a n,2 1 (55) Thus it is sufficient to show that that is Note that [M n (1 + β) (δ 1 + β)f n ] a n,1 a n,2 [M n(1 + β) (α + β)f n ] an,1 a n,2, n j + 1 (56) 1 δ 1 an,1 a n,2 [M n(1 + β) (α + β)f n ](1 δ 1 ), n j + 1 (57) [M n (1 + β) (δ 1 + β)f n ]() an,1 a n,2 Consequently, we need only to prove that or equivalently (), n j + 1 (58) () [M n(1 + β) (α + β)f n ](1 δ 1 ) [M n (1 + β) (δ 1 + β)f n ](), (59) δ 1 [M n(1 + β) (α + β)f n ] 2 [M n (1 + β) βf n ]() 2 2 F n () 2 (n) (510) Since (n) is an increasing function of n(n j + 1), letting n j + 1 in (510) we obtain δ 1 (j + 1) [M j+1(1 + β) (α + β)f j+1 ] 2 [M j+1 (1 + β) βf j+1 ]() 2 [M j+1 (1 + β) (α + β)f j+1 ] 2 F j+1 () 2 (511) which proves the main assertion of Theorem 51 The result is sharp for the functions defined by (24)

8 N Magesh et al / Uniformly convex functions 25 Theorem 52 Let the function f i (z)(i 1, 2) defined by (51) be in the class T S(λ, α, β, j) If the sequence {} is non-decreasing Then the function belongs to the class T S(λ, δ 2, β, j) where h(z) z (a 2 n,1 + a 2 n,2)z n (512) δ 2 [M n(1 + β) (α + β)f n ] 2 2[M n (1 + β) βf n ]() 2 2 2F n () 2 Proof By virtue of Theorem 22, we have for f j (z)(j 1, 2) T S(λ, α, β, j) we have and [ [Mn (1 + β) (α + β)f n ] [ [Mn (1 + β) (α + β)f n ] ] 2 a 2 n,1 ] 2 a 2 n,2 [ ] 2 [Mn (1 + β) (α + β)f n ] a n,1 1 [ ] 2 [Mn (1 + β) (α + β)f n ] a n,2 1 (513) (514) It follows from (513) and (514) that 1 2 [ [Mn (1 + β) (α + β)f n ] Therefore we need to find the largest δ 2, such that that is [M n (1 + β) (δ 2 + β)f n ] 1 δ ] 2 (a 2 n,1 + a 2 n,2) 1 (515) [ ] 2 [Mn (1 + β) (α + β)f n ], n j + 1 δ 2 [M n(1 + β) (α + β)f n ] 2 2[M n (1 + β) βf n ]() 2 2 2F n () 2 Ψ(n) Since Ψ(n) is an increasing function of n, (n j + 1), we readily have which completes the proof δ 2 Ψ(j + 1) [M j+1(1 + β) (α + β)f j+1 ] 2 2[M j+1 (1 + β) βf j+1 ]() 2 [M j+1 (1 + β) (α + β)f j+1 ] 2 2F j+1 () 2 6 Acknowledgement The authors are very much thankful to the referee for his/her useful suggestions and comments References [1] HS Al-Amiri, On a subckass of close-to-convex functions with negative coefficients, Mathematics, Tome, 31(1)(54)(1989), 1 7 [2] R Bharati, R Parvatham, and ASwaminathan, On subclasses of uniformly convex functions and corresponding class of starlike functions, Tamkang J Math, 28(1)(1997), [3] BA Frasin, On the analytic functions with negative coefficients, Soochow J Math, 31(3)(2005),

9 26 N Magesh et al / Uniformly convex functions [4] BA Frasin, Some applications of fractional calculus operators to certain subclass of analytic functions with negative coefficients, Acta Universitatis Apulensis, 23(2010), [5] AW Goodman, On uniformly convex functions, Ann polon Math, 56(1991), [6] AW Goodman, On uniformly starlike functions, J Math Anal Appl, 155(1991), [7] VP Gupta and PK Jain, Certain classes univalent analytic functions with negative coefficients II, Bull Austral Math Soc, 15(1976), [8] W Ma and D Minda, Uniformly convex functions, Ann Polon Math, 57(1992), [9] N Magesh, On certain subclasses of analytic and bi-univalent functions, Preprint [10] G Murugusundaramoorthy and N Magesh, On certain subclasses of analytic functions associated with hypergeometric functions, Applied Mathematics Letters, 24(2011), [11] F Running, Uniformly convex functions and a corresponding class of starlike functions, Proc Amer Math Soc, 118(1993), [12] T Rosy, Study on subclasses of univalent functions, Thesis, University of Madras, 2002 [13] S M Sarangi and BA Uralegaddi, The radius of convexity and starlikeness of certain classes of analytic functions with negative coefficients I, Atti Accad Naz Lincei rend Cl Sci Fis Mat Natur, 65(8)(1978), [14] H Silverman, Univalent functions with negative coefficients, Proc Amer Math Soc, 51(1975), [15] H M Srivastava, S Owa and S K Chatterjea, A note on certain classes of starlike func- tions, Rend Sem Mat Univ Padova, 77(1987), [16] BA Stephand and KG Subramanian, On a subclass of Noshiro-Type analytic functions with negative coefficients, Conference Proc Ramanujan Mathematical Society, 1998 [17] KG Subramanian, TV Sudharsan, P Balasubrahmanyam and H Silverman, Class of uniformly starlike functions, Publ Math Debercen, 53(4)(1998), [18] KG Subramanian, G Murugusundaramoorthy, P Balasubrahmanyam and H Silverman, Subclasses of uniformly convex and uniformly starlike functions, Math Japonica, 42(3)(1995), Received: October 17, 2012; Accepted: November 30, 2012 UNIVERSITY PRESS