STARLIKE AND CONVEX FUNCTIONS OF POSITIVE ...

STARLIKE AND CONVEX FUNCTIONS OF POSITIVE ORDER A.K.MISHRA Abstract. In the present paper we discuss (without proof) about two classical problems on star like and convex functions of positive order. The first problem deals with nontrivial inclusion relation between the classes of convex functions of order α and star like functions of order β. The second problem is about coefficient estimate problem for the inverse function of a starlike function of order α.

1. Univalent Starlike and Convex Functions Let A be the class of functions analytic in the open unit disk U := {z : z ∈ C and |z| < 1}. The family A is a locally convex linear topological space under the metrizable topology of uniform convergence on compact subsets of U (cf. [37] see also [11]). Let A1 denote the class of functions in A and represented by the normalized series f (z) = z +

∞ X

an z n ,

(z ∈ U).

n=2

As usual, we shall denote by S the class of univalent functions in A1 . It is well known that S is a compact subset of A. A function f ∈ A1 is called starlike (respectively convex ) if f maps U univalently on to a region starlike with respect to the origin (a convex region). The classes of univalent starlike and convex functions are denoted here respectively by S ∗ and CV. Since a convex region containing the origin is also a star like region with respect to origin we have (1.1)

CV ⊂ S ∗ .

It follows from geometric considerations that f is a univalent starlike map if and only if for every 0 < r < 1, ∂ argf (reiθ ) > 0 (0 < θ < 2π) ∂θ 2000 Mathematics Subject Classification. 30C45, 30C80. Key words and phrases. Univalent function, p−valent function, Starlike function of order α, Convex function of order α Differential subordination, Inverse function, Coefficient estimate . 1

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A.K.MISHRA

and f is a univalent convex function if and only if zf 0 (z) is a starlike function. The transformation A : A1 → A1 given by (1.2)

A(f )(z) = zf 0 (z)

is a linear homeomorphism from CV onto S ∗ and is popularly known as Alexander’s transformation. The function f ∈ A1 is said to be in the class of starlike functions of order α, (0 ≤ α < 1), denoted by S ∗ (α), if the rate of change of argf (reiθ ) with respect to θ is not less than or equal to α, i.e. ∂ argf (reiθ ) > α (0 < θ < 2π). ∂θ The class of univalent convex functions of order α i.e. CV(α) is defined analogously via the Alexander’s transformation. More over, (1.3)

CV(α) ⊂ S ∗ (α).

The following refinement of the inclusion relation (1.1) is also well known [7]: 1 ∗ (1.4) CV ⊂ S . 2 If R(α) denotes the class of functions f ∈ A1 satisfying f (z) < >α z then we also know that (1.5)

1 . CV ⊂ R 2

The inclusion relations (1.4) and (1.5) are proper and these two together are popularly known as the Strohh¨acker-Marx result. However, it is a consequence of the Herglotz theorem that (Cf. [3, 4]) 1 1 ∗ (1.6) H(CV) = H S =R 2 2 where H denotes the closed convex hull. For a given α, (0 ≤ α < 1), the problem of determining sharp value of β := β(α) such that (1.7)

CV(α) ⊂ S ∗ (β),

has been completely resolved in [38]. Thus, if 0 ≤ α < 1 and f ∈ CV(α) then f ∈ S ∗ (β) where   1−2α [1 − 2(2α−1) ] α = 6 21 22(1−α) β=  1 α = 12 2log2

STARLIKE AND CONVEX FUCTIONS...

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Discussions on the classes S ∗ , CV, S ∗ (α) and CV(α) can be found for example in [7, 11, 32, 33]. 2. p−valent Starlike and Convex Functions A function f ∈ A is said to be p-valent (p ∈ N := {1, 2, · · · }) in U if for every complex number w the equation f (z) = w has no more than p-solutions in U and there is a complex number w0 such that f (z) = w0 has exactly p-solutions in U. Now, let q ∈ N and let Aq denote the subclass of A consisting of functions of the form ∞ X

q

f (z) = z +

an z n ,

(z ∈ U).

n=q+1

A function f ∈ Aq is said to be p−valent convex [9], p ∈ N, p ≥ q if there exists a positive number ρ = ρ(f ), 0 < ρ < 1, such that zf 00 (z) < 1+ 0 > 0 (z ∈ U) f (z) and

2π

zf 00 (z) dθ = 2πp < 1+ 0 f (z) 0 for ρ < |z| < 1 and z = reiθ . We denote the class of p−valent convex functions by CV(p, q). A function f ∈ Aq is said to be in the class of p−valent starlike functions , (p ∈ N, p ≥ q), Rz denoted here by S ∗ (p, q), if q 0 (f (t)/t)dt is in CV(p, q) ( See [9]). Z

In contrast to the univalent case the class CV(p, q) is not a subset of S ∗ (p, q). However, CV(p, p) ⊂ S ∗ (p, p). While a function f is in S ∗ (p, p) if and only if f (z) = (g(z))p for some univalent star like function g; every function in CV(p, p) is not necessarily the pth power of some univalent convex function. For suitable examples see e.g. [9]. The analogous definitions for p−valent star like functions of order α and p−valent convex functions of order α were developed by Kapoor and the present author, (See [16]). Thus a function f ∈ Aq is said to be p−valent star like of order α, (p ∈ N, p ≥ q) if there exists a positive number ρ = ρ(f ), 0 < ρ < 1, such that 0 zf (z) < > pα (z ∈ U) f (z) and

2π

zf 0 (z) < dθ = 2πp f (z) 0 for ρ < |z| < 1 and z = reiθ . We denote the class of p−valent star like functions of order α by S ∗ (p, q; α). Similarly, a function f ∈ Aq is said to be p−valent convex of order α, Z

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A.K.MISHRA 0

(p ∈ N, p ≥ q) if the function zfq is a member of S ∗ (p, q; α). We denote the class of p−valent convex functions of order α by CV(p, q; α). Again it is well known [10] that , in contrast to the univalent case, the analogoues of the Strohh¨acker-Marx results are not true in the multivalent case. In fact, there does not exist a β > 0 such that CV(p, p) is a subset of S ∗ (p, p; β). Similarly, there does not exist a γ > 0 such that for every function in CV(p, p) the relation f (z) < >γ zp would hold. Kapoor and the present author [18] proved the following: Theorem 2.1. Let f, given by the series ∞ X p f (z) = z − |ak+p |z k+p

(z ∈ U),

k=1

be a member of CV(p, p; α), 0 ≤ α < 1. Then (i) f is in S ∗ (p, p; β) where β := β(α) =

p+1 2p + 1 − pα

(ii)
γ

for every z ∈ U, where p2 (1 − α) + p + 1 (p + 1)(p + 1 − α) The bounds on β(α), and γ(α) are sharp. γ = γ(α) =

In this context the following recent results is now well known. This result appears as particular case of certain general results by Patel etal. proved by using techniques of differential subordination. (See e.g. Patel and Mishra [30]). Theorem 2.2. Suppose that (2.1)

1 2

1 1− ≤ α < 1, p

then CV(p, p; α) ⊂ S ∗ (p, p; β) where β := β(α) = The bound on β is the best possible.

2 F1

1 1, 2(p − α); p + 1; 2

−1


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Problem 2.3. Find the sharp value of α0 > 0 such that for each α0 ≤ α < 1 there exists a β(α) > α and CV(p, p; α) ⊂ S ∗ (β) 3. Meromorphic Starlike and Convex Fuctions A function f analytic in the punctured unit disc U \ {0} and having a simple pole with residue 1 at z = 0 is said to be univalent meromorphic starlike (respectively, univalent meromorphic convex ) if f maps U \ {0} on to a region whose complement is a star like region with respect to origin (respectively, convex region). Therefore, f is univalent meromorphic starlike if and only if −

∂ argf (reiθ ) > 0 (0 < θ < 2π) ∂θ

and f is a univalent meromorphic convex function if and only if −zf 0 (z) is a univalent meromorphic starlike function. We denote the classes of univalent meromorphic starlike P ∗ P functions and univalent meromorphic convex functions by S and CV respectively. Again from geometric consideration we have X X CV ⊂ S ∗. The concepts of univalent meromorphic starlike functions of order α and univalent meromorphic convex function of order α, 0 ≤ α < 1 have been developed in literature in analP ∗ P ogous manner. Let S (α) and CV(α) denote the classes of univalent meromorphic starlike functions of order α and univalent meromorphic convex functions order α respecP P ∗ tively. Inclusion relations between CV(α) and S (β) analogous to (1.7) seems to be not known in the literature. Techniques of differential subordination adopted by Patel and Mishra for the proof of Theorem2.2 does not work here. P Problem 3.1. Does there exist a nontrivial inclusion relation between CV(α) and P ∗ S (β) analogous to (1.7)? 4. Coefficient Estimates for Inverse Functions of Univalent Star like Functions of Positive Order It is well known that every function f ∈ S has an inverse f −1 , defined by f −1 (f (z)) = z,

(z ∈ U)

and f (f

−1

(w)) = w,

1 . |w| < r0 (f ), r0 (f ) ≥ 4

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A.K.MISHRA

In fact, the inverse function f −1 is analytic in |w| < ro (f ) and if f is given by the series (4.1)

f (z) = z +

∞ X

an z n

(z ∈ U)

n=2

then f −1 (w) = w − a2 w2 + (2a22 − a3 )w3 − (5a32 − 5a2 a3 + a4 )w4 + . . . . (4.2)

=w+

∞ X

An wn

(say).

n=2

Let S −1 be the class of inverse functions f −1 of functions f ∈ S. The classes S ∗−1 and S ∗ (α)−1 are analogously defined. Loewner, using his parametric method (cf.[26]; also see [12], p.222) proved that if f −1 , given by (4.2), is in the class S −1 or S ∗−1 , then the sharp estimate Γ(2n + 1) |An | ≤ , n = 2, 3, . . . Γ(n + 2)Γ(n + 1) holds, the inverse of the Koebe function i.e. K0−1 ; being the extremal function for all n. The above coefficient estimate problem for the classes S −1 and S ∗−1 is also investigated by the methods developed by Shaeffer and Spencer [34] , FitzGerald [8], Baernstein [2], Poole [31] and others. Prior to de-Branges result on the sharp coefficient bounds for the whole class S, the coefficient estimate problem was established for several subclasses of S, e.g. the classes of convex functions, starlike functions of order α, close-to-convex functions, normalized integrals of functions with positive real part etc. However, in contrast, although for the class of inverse functions the coefficient problem for the whole classes S −1 and S ∗−1 had been completely solved as early as in 1923 [26], only few complete results are known on the coefficient estimates for the inverses of functions in the above subclasses [15, 21, 25, 35]. In certain cases, the coefficients of inverse functions of the functions in some of these subclasses show unexpected behaviour. For example, it is known that if f is a univalent convex function and f −1 is given by (4.2), then |An | ≤ 1 for n = 1, . . . , 8 [5, 24] and z ; while |A10 | > 1 [20] and the equality holds for the inverse of the function K1/2 (z) = 1−z exact bounds on |A9 | and |An | for n > 10 are still unknown. Krzyz, Libera and Zlotkiewicz [22] showed that if f −1 , given by (4.2), is in S ∗ (α)−1 then (4.3)

|A2 | ≤ 2(1 − α)

and (4.4)

 (1 − α)(5 − 6α), 0 ≤ α ≤ |A3 | ≤ (1 − α), 2 ≤ α < 1 3

2 3


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The estimate (4.3) and the first estimate in (4.4) are sharp for the function Kα−1 while the −1 second estimate in (4.4) is sharp for the function Kα,2 , where p z n (4.5) Kα (z) = and K (z) = Kα (z n ), n = 2, 3, . . . α,n 2(1−α) (1 − z) The determination of sharp estimates on |An | for n ≥ 4 is hitherto an open problem for the class S ∗ (α)−1 . Kapoor and the present author[19] proved the following: Theorem 4.1. Let f ∈ S ∗ (α), 0 ≤ α < 1 and for |w| < 14 , (4.6)

f

−1

(w) = w +

∞ X

An wn .

n=2

Then, (a) for α ∈ [0, n2 ), |An | ≤

(4.7) (b) for α ∈ (4.8)

h

k k+1 n n

Γ(2n(1 − α) + 1) Γ(n + 1)Γ(2n(1 − α) + 2 − n)

, k = 2, . . . , n − 2, |An | ≤

Γ(2n(1 − α) + 1) n(n − 1)Γ(n − k)Γ(2n(1 − α) + 1 + k − n)

(c) for α ∈ [1 − n1 , 1) (4.9)

|An | ≤

2(1 − α) . n−1

The estimates (4.7) and (4.9) are sharp. We observe that the bounds on |An | found in the above Theorem4.1 are sharp when α lies in small neighborhoods of the end points of the interval [0, 1) leaving large gap where sharpness of the estimates have not been established. Problem 4.2. Find sharp estimates and respective extremal functions for |An | for f ∈ ), k = 2, . . . , n − 2. S ∗ (α), α ∈ [ nk k+1 n Let A1 (m), (m ∈ N), be the class of functions f analytic in the open unit disc U and having m-fold symmetric series expansion ! ∞ X (4.10) f (z) = z 1 + avm z vm . v=1

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A.K.MISHRA

The function f ∈ A1 (m) is said to be in S ∗ (m, α, β), (0 ≤ α < 1, 0 < β ≤ 1); the class of univalent starlike functions of order α and type β in U, if the condition (4.11)

|h(z) − 1|