On Multivalent Functions Associated with Fixed Second Coefficient and the Principle of Subordination

Transcription

1 International Journal of Mathematical Analysis Vol. 9, 015, no. 18, HIKARI Ltd, On Multivalent Functions Associated with Fixed Second Coefficient and the Principle of Subordination C. Selvaraj Department of Mathematics Presidency College (Autonomous) Chennai , India S. Stelin Department of Mathematics Tagore Engineering College, Vandalur Chennai , India Copyright 014 C. Selvaraj and S. Stelin. 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. Abstract In this paper we obtain certain results on multivalent functions, as an application of the principle of subordination, with fixed second coefficient. The influence of the second coefficient of p-valent functions is realized in the results obtained. Mathematics Subject Classification: primary 30C80, secondary 30C45 Keywords: convex p-valent, starlike p-valent, fixed second coefficient, fixed initial coefficient, differential subordination

2 884 C. Selvaraj and S. Stelin 1 Introduction and Known sults Let A p,n denote the class of analytic functions defined on the unit disc of the form E = {z C : z < 1} f(z) = z p + a n+p z n+p + a n+p+1 z n+p and let A p = A 1,p, the class of p-valent functions and A 1,1 := A 1 = A the class of all normalized regular functions. Let S p(α), C p (α), respectively denote the classes of all p-valent starlike and p-valent convex functions of order α, (0 α < 1), satisfying the inequalities and 1 zf (z) p f(z) 1 (zf (z)) p f (z) > α, (z E) > α, (z E). Clearly S 1 = S and C 1 (0) = C, the classes of starlike and convex functions and for p = 1 and α 0 they represent the classes of starlike and convex functions of order α, respectively. In [,3,4,5,6,7,8,9,10,15,17], various authors have studied the classical Gronwall problem and obtained results on coefficient, distortion, growth and covering problems for various subclasses of univalent functions and these results were shown to be improvements over the classical results because of the fixed second coefficient. In [1,13] Ali et al and Nagpal and Ravichandran applied the modified theory of differential subordination to univalent function theory to obtain results that were shown to be improvements over earlier results. In this paper, we extend their ideas to p-valent function theory and obtain various results. Now we state below some basic definitions and important theorems which are required in the sequel. Definition 1.1. [1] Let f and F be analytic in E. Then f is said to be subordinate to F, written f F, if there is a function w(z) analytic in E, w(0) = 0, w(z) < 1 such that f(z) = F (w(z)). However, if F is univalent in E, then f is subordinate to F, if f(0) = F (0) and f(e) F (E). Definition 1.. [11] Let Q be the class of functions q that are analytic and injective in E U(q), where U(q) := {ζ E lim z ζ q(z) = }, and are such that q (ζ) 0 for ζ in E \ U(q). Definition 1.3. [1] Let Ω be a domain in C, n N and β > 0. Let q Q be such that q (0) β. The class Ψ n,β (Ω, q) consists of β-admissible functions ψ : C C C satisfying the following conditions:

3 On multivalent functions 885 (i) ψ(r, s) is continuous in a domain G C C, (ii) (q(0), 0) G and ψ (q(0), 0) Ω, (iii) ψ (q(ζ), mζq (ζ)) / Ω whenever (q(ζ), mζq (ζ)) G, ζ E \ U(q) and m n + q (0) β q (0) + β. We write Ψ 1,β (Ω, q) as Ψ β (Ω, q). Theorem 1.1. [1] Let q Q, q(0) = a, β > 0 with q (0) β, and ψ Ψ n,β (Ω, q) with associated domain G. Let H β [a, n] consist of all analytic functions h of the form h(z) = a + βz n + p n+1 z n , β C is fixed, h H β [a, n]. If (h(z), zh (z)) G for z E and ψ (h(z), zh (z)) Ω, (z E), then h q. The special case of Ω being the half-plane = {w : w > 0} is important for our discussion. The function q(z) = a+āz, z E, where a > 0, 1 z is univalent in E \ {1} and satisfies q(e) =, q(0) = a and q Q. Let Ψ n,β {Ω, a} := Ψ n,β {a}. Set Ψ β {a} := Ψ 1,β {a}. The class Ψ n,β {Ω, a} consists of those functions ψ : C C that are continuous in a domain G C C with (a, 0) G and ψ(0, a) Ω, and satisfy the admissibility condition: and ψ(iρ, σ) / Ω whenever (iρ, σ) G σ 1 ( n + a β ) a iρ a + β a (1.1) where ρ R and n 1. If a = 1, then (1.1) simplifies to ψ(iρ, σ) / Ω whenever (iρ, σ) G and σ 1 ( n + β ) (1 + ρ ) (1.) + β where ρ R, and n 1. In this particular case theorem 1.1 becomes Theorem 1.. [1] Let h H β [a, n] with a > 0 and 0 < β a. (i) Let ψ Ψ n,β (Ω, a) with associated domain G. If (h(z), zh (z)) G and ψ (h(z), zh (z)) Ω, (z E), then h(z) > 0 (z E). (ii) Let ψ Ψ n,β {a} with associated domain G. If (h(z), zh (z)) G and ψ (h(z), zh (z)) > 0, (z E), then h(z) > 0 (z E).

4 886 C. Selvaraj and S. Stelin One of the classical Marx-Strohhäcker results claims that every convex function is a starlike function of order 1. That is, if f A, then ( ) f (z) + 1 > 0, (z E) implies. In fact, ( ) zf (z) > 1, (z E) f(z) Theorem 1.3. [Marx-Strohhäcker, 1, Theorem.6a] If f A, then the following implications hold: ( ) ( ) zf f (z) + 1 (z) > 0 = > 1 f(z) f (z) > 1 Main sults = f(z) z > 1 In this section we extend the results of Ali et al to prove Marx-Strohhäcker like results to p-valent functions. ( We need ) to find a domain G containing the right half plane so that G (z E) implies zf (z) > p, p f (z) f(z) (z E). The domain G cannot be taken as the half plane {w C : w > α}, with α < 0, for functions in A p. However, it is possible to take such a G for functions in A p,b where f A p,b is of the form with b = a p+1 fixed. f(z) = z p + bz p+1 + a p+ z p (z E) Theorem.1. If f A p,b with b 1, then 1 ( ) p f (z) + 1 > b 1, (z E) p( b + 1) implies ( ) zf (z) > p, (z E). f(z)

5 On multivalent functions 887 Proof. Define the function h : E C by h(z) = zf (z) pf(z) 1 (.1) Then, h(z) = 1 + bz + (a p p p+ b ) z +... belongs to H b [1, 1]. Also (.1) p implies h(z) + 1 = zf (z) pf(z) Equivalently, Differentiation yields: p[h(z) + 1]f(z) = zf (z) p[h (z)]f(z) + pf (z)[h(z) + 1] = [ + f (z)] Dividing by pf (z), h(z) f(z)h (z) f (z) = 1 p [ ] f (z) + 1 This implies or That is h(z) + 1 h(z) zh (z) z f (z) f(z) = 1 p [ ] f (z) zh (z) p[h(z) + 1] α = 1 [ ] p f (z) + 1 α. ( ) 1 p f (z) + 1 α = ψ (h(z), zh (z)) (.) where ψ(r, s) := r+1 + s α. If we set p(r+1) α := b 1 p( b + 1) (.3) then α satisfies 1 α 0. p The function ψ : C\{1} C C is continuous in the domain G = C\{ 1} C. [ ] Note that (1, 0) G and ψ(1, 0) = 1 α > 0 since p N and α 1, 0. p Now we need to verify the admissibility condition (1.).

6 888 C. Selvaraj and S. Stelin We have ψ(iρ, σ) = iρ+1 + σ (1 iρ) α and p(1+ρ ) ψ(iρ, σ) = 1 + σ p(1 + ρ ) α 1 1 ( p + b ) p + b = b 1 p( b + 1) α =0 ( ) whenever ρ R and σ 1 p + β (1 + ρ ), β = b. Thus ψ +β Ψ p, b {1}. Using the hypothesis together with (.) yields α ψ (h(z), zh (z)) > 0, (z E). Now applying (ii) of theorem 1., the conclusion namely, follows. zf (z) f(z) > p, (z E) Corollary.. For p = 1, we obtain the results of [13, Theorem.]. Corollary.3. For p = 1, b = 1 Theorem.1 reduces to [1, Theorem.6a]. Corollary.4. For p = 1, b = 0, we get α = 1. Therefore, in this case, this theorem reduces to [1, Theorem.6i]. Theorem.5. If f A p,b with b 1, then 1 ( ) p f (z) + 1 > b 1, (z E) p( b + 1) implies f (z) pz > 1 p 1, (z E). where the square root is so chosen as to satisfy 1 = 1. Proof. Take h(z) = f (z) 1 pzp 1 (.4)

7 On multivalent functions 889 Then h is a function from E C and [ b(p + 1) (p + )ap+ h(z) = 1 + z + p p 1 b (p + 1) 4 p ] z +... Clearly h is analytic in E and h H b(p+1) [1, 1]. Also (.4) implies p Equivalently, (h(z) + 1) = 4 f (z) pz p 1. (h(z) + 1) pz p 1 = 4f (z). Differentiating and simplifying we obtain: (p 1) + pzp (h(z) + 1) h (z) f (z) = f (z). Therefore 1 p (( f (z) ) where ψ(r, s) := 1 + ) + 1 α = 1 + zh (z) p[h(z) + 1] α = ψ (h(z), zh (z)), (z E) s α, and p(r+1) (.5) α := b 1 p[ b + 1] (.6) The function ψ : G = C \ { 1} C C is continuous, (1, 0) G, and ψ(1, 0) = 1 α > 0, since α [ 1, 0]. Now we need to verify the p admissibility condition: We have whenever ρ R and σ 1 σ ψ(iρ, σ) =1 + (1 iρ) α p(1 + ρ ) σ ψ(iρ, σ) =1 + p(1 + ρ ) α 1 1 ( p + b ) α p + b = b 1 p( b + 1) α =0 ( ) p + β (1 + ρ ), β = b. Thus ψ +β

8 890 C. Selvaraj and S. Stelin Ψ p, b {1}. Using the hypothesis together with (.5) yields ψ (h(z), zh (z)) > 0, (z E). Therefore, by (ii) of Theorem 1., it follows that h(z) > 0, z E. Equivalently, f (z) pz p 1 > 1. Corollary.6. For p = 1, Theorem.5 reduces to the results obtained in [13, Theorem.4]. Corollary.7. For p = 1 and b = 1, α = 0 and in this case the theorem reduces to [1, Theorem.6a]. Theorem.8. If f A p,b with b 1 then, implies Proof. Take zf (z) pf(z) (p 1) + (p + 1) b >, (z E) p(1 + b ) f(z) > 1, (z E). z p h(z) := f(z) z p 1 (.7) Then h is a function from E C and h(z) = 1 + bz + a p+ z +... is analytic in E since h H b [1, 1]. Now (.7) implies Differentiating and simplifying we get (h(z) + 1) z p = f(z). pf(z) + z p+1 h (z) = zf (z). This implies Therefore, 1 + zh (z) p[h(z) + 1] = zf (z) pf(z). zf (z) pf(z) α = 1 + zh (z) p[h(z) + 1] α = ψ (h(z), zh (z)) (.8)

9 On multivalent functions 891 where ψ(r, s) := 1+ s α, is continuous in the domain G := C { 1} C p(r+1) and (1, 0) G. If we set (p 1) + (p + 1) b α :=, p( b + 1) [ ] p 1 then α, 1 and ψ(1, 0) = 1 α > 0. It remains to verify the p admissibility condition. We have σ ψ(iρ, σ) =1 + (1 iρ) α p(1 + ρ ) σ ψ(iρ, σ) =1 + p(1 + ρ ) α 1 1 ( p + b ) α p + b (p 1) + (p + 1) b = α p( b + 1) =0 ( ) whenever ρ R and σ 1 p + β (1 + ρ ), β = b. Thus ψ +β Ψ p, b {1}. Applying (.8) together with the hypothesis we obtain ψ (h(z), zh (z)) > 0, (z E). Therefore, by (ii) of Theorem 1., we conclude that h(z) > 0, z E. Equivalently, f(z) > 1, (z E). z p Corollary.9. For p = 1, Theorem.8 reduces to the results obtained in [13, Theorem.4]. Corollary.10. For p = 1 and b = 1, α = 0 and in this case the theorem reduces to [1, Theorem.6a]. Theorem.11. If f A p,b is locally p-valent with b 1, then the following implication holds: f (z) p(1 + b ) pz > (z E) p 1 8

10 89 C. Selvaraj and S. Stelin implies f(z) z p > 1 (z E), where the square root is chosen so as to satisfy 1 = 1. Proof. Put h(z) = f(z) z p 1. (.9) Then h(z) is an analytic function: E C and is of the form h(z) = 1 + bz Since f A p,b and is analytic in E, it follows that h H b [1, 1]. p(1+ b ) If we set α :=, then p 8 α p and f (z) pz α = zh (z) + ph(z) + p α = ψ (h(z), zh (z)) (z E) (.10) p 1 p where ψ(r, s) := pr+s+p p α. The function ψ is continuous in the domain [ p ]. G = C C, (1, 0) G and ψ(1, 0) = 1 α > 0 since α Now we need to verify the admissibility condition: ipρ + σ + p (iρ, σ) = α p if ρ R and σ 1 0 ( p + β ) (1 + ρ ), β = b. + β, p If we let ζ = ξ + iη = p+σ+ipρ, then upon using the conditions on ρ and σ, we p get ξ = p + σ 1 p 1 ( p + β ) (1 + ρ ) p + β = 1 1 ( p + 1 b ) (1 + 4η ) p 1 + b (p 1) + p b 4η =. p(1 + b )

11 On multivalent functions 893 This implies η p(1 + b ) p(1 + b ) [ ξ ] (p 1) + p b p(1 + b ) and this in turn would mean that ζ lies in the interior of the parabola [ ] η = ξ. Now, A simple computation shows that (p 1) + p b p(1 + b ) ζ = ξ + ξ ξ + iη = + η. ξ + η { [p(1 + b ) 1] + 4η } (1 + 4η ) [p(1 + b )] and upon using the inequality ab a+b we get This implies ξ + ξ + η and hence so that ξ + η 1 { [p(1 + b ) 1] η }. 4p(1 + b ) { } [p(1 + b ) 1] η + 8p(1 + b ) p(1 + b ) = 8 ζ = ξ + ξ + η = ζ α 0, p(1 + b ) (p 1) + p b 4η 4p(1 + b ) which is exactly the admissibility condition. Thus ψ Ψ b {1}. From the hypothesis and (.10), we obtain ψ (h(z), zh (z)) > 0 (z E). Therefore, by applying (ii) of theorem 1. we conclude that h(z) > 0 (z E). 8 Equivalently, f(z) z p > 1 (z E).

12 894 C. Selvaraj and S. Stelin Corollary.1. For p = 1, this theorem reduces to the result obtained in [13, Theorem.8]. Corollary.13. If p = 1 and b = 1 then α = 1. In this case this theorem reduces to [1, Theorem.6a] Acknowledgements: The first author acknowledges the support received from Tamil Nadu State Council for Higher Education through its Minor search Programme for Government Arts and Science Colleges. ferences [1] R. M. Ali, S. Nagpal and V. Ravichandran, Second order differential subordination for analytic functions with fixed initial coefficient, Bull. Malays. Math. Sci. Soc.() 34 (011), [] O. P. Ahuja, The influence of second coefficient on spirallike and the Robertson functions, Yokhama Math. J. 34 (1-)(1986) [3] H. S. Al-Amiri, On p-close-to-star functions of order α, Proc. Amer. Math. Soc. 9 (1971) [4] V. V. Anh, Starlike functions with a fixed coefficient, Bull. Austral. Math. Sco. 39 (1) (1989) [5] M. K. Aouf, Bounded spiral-like functions with fixed second coefficient, Internat. J. Math. Sci. 1(1) (1989) [6] M. Finkelstein, Growth estimates of convex functions, Proc. Amer. Math. Soc. 18(1967) [7] R. M. Goel, A class of univalent functions with fixed second coefficients, J. Math. Sci. 4(1969) [8] R. M. Goel, The radius of convexity and starlikeness for ertain classes of analytic fnctions with fixed second coefficients, Ann. Univ. Mariarcurie sklodowoska sect. A 5 (1971) [9] T. H. Grownwall, On the distortion in conformal mapping when the second coefficient in the mapping functions has an assigned value, Proc. Nat. Acad. Sci. U.S.A. 6 (190)

13 On multivalent functions 895 [10] V. Kumar, On univalent functions with fixed second coefficient, Indian J. Pure Appl. Math. 14 (11) (1983) [11] S. S. Miller and P. T. Mocanu, Differential subordinations and univalent functions, Michigan Math. J. 8 (1981), [1] S. S. Miller and P. T. Mocanu, Differential subordinations: Theory and Applications, Dekker, New York, 000. [13] S. Nagpal and V. Ravichandran, Applications of theory of differential subordination for functions with fixed initial coefficient to univalent functions, Anna. Poloni. Math (01), [14] S. Ozaki, On the theory of multivalent functions II, Sci. p. Tokyo Bunrika Daigaku Sect. A 4 (1941), [15] K. S. Padmanabhan, Estimamtes of growth for certain convex and closeto-convex functions in the unit disc, J. Indian Math. Soc. (N.S) 33(1969), [16] R. Singh and S. Singh, Some sufficient conditions for univalence and starlikeness, Colloq. Math. Vol. XLVII (198) [17] P. D. Tuan, V. V. Anh, Extremal problems for functions of positive real part with a fixed coefficient and applications, Czechoslovak Math. J. 30 (105)(1980) ceived: December 15, 014; Published: March 3, 015