On a Simple Sufficient Condition for the Uniform Starlikeness

The aim of this paper is to prove the theorem which generates many examples of functions belonging to a geometrically defined class of uniformly starlike functions introduced by Goodman in 1991. Only a very few explicit uniformly starlike functions were known until now. Next we obtain inclusion relations between some subclasses of convex functions and the class of uniformly starlike functions.


Introduction
Let S denote the class of all functions f that are analytic and univalent in the open unit disk U = {z ∈ C : |z| < 1} and normalized by f (0) = f (0) − 1 = 0.
A set D ⊂ C is said to be starlike with respect to w 0 , an interior point of D, if the intersection of each half-line beginning at w 0 with the interior of D is connected.
We denote by ST the class of all starlike functions, i.e., the subclass of S consisting of functions that map U onto domains starlike with respect to w 0 = 0 (briefly starlike domains). Recall that a function f ∈ S is starlike if and only if Communicated by See Keong Lee.
B Agnieszka Wiśniowska-Wajnryb agawis@prz.edu.pl Let CV denote the class of all functions f ∈ S that are convex in U , i.e., such that f (U ) is a convex domain.
Let γ : z = z(t), t ∈ [a, b], be a smooth, directed arc and suppose that a function f is analytic on γ . Then the arc f (γ ) is said to be -starlike with respect to -convex if the argument of the tangent to f (γ ) is a nondecreasing function of t.
Recall that a function f ∈ S is in the class UCV (UST ) if for every circular arc γ ⊂ U with center ζ ∈ U , the arc f (γ ) is convex (starlike with respect to f (ζ )).
If we take ζ such that |ζ | ≤ k, 0 ≤ k ≤ 1 we obtain the following natural extension of the concept of uniform starlikeness (see [11] and also [10,12]). Namely, a function f ∈ S is said to be k-uniformly starlike in U , if the image of every circular arc γ contained in U with center at ζ , where |ζ | ≤ k, is starlike with respect to f (ζ ).
We denote by k-UST the class of all k-uniformly starlike functions. Notice that 0-UST = ST and 1-UST = UST . Moreover, it is clear that Goodman obtained the analytic conditions for UCV and UST expressed by two complex variables. For an arbitrary k such that 0 ≤ k ≤ 1, the class k-UST can be characterized (see [11]) as follows For k = 1, we get Goodman's condition for uniform starlikeness. It turned out that (see [3,7]) Finding this analytic condition essentially simplified further investigations of uniformly convex functions ( [3,4,7,8]).
It is more difficult to investigate the class UST (k-UST ) because of its characterization in terms of two complex variables. In particular checking whether a function belongs to the class UST leads to very complicated computations. Hence only simple examples of uniformly starlike functions are known.
In this paper, we give many examples of members of the class of uniformly starlike functions UST and we establish its connections with some subclasses of convex functions. For instance, we get that all functions convex of order 3/4 are uniformly starlike. Goodman ([2]) gave some examples of uniformly starlike functions. He proved that

Some Members of the Class of Uniformly Starlike Functions
It was proven in [11] that For k = 1 we get This result was mentioned by Goodman, but without a proof. We have also more general result (see [11]): If 0 < k ≤ 1 and for some integer n ≥ 2 For k = 1 we get the result of Merkes and Salmassi ( [5]), which improves the bound |A| ≤ 1/( √ 2n) obtained by Goodman. Merkes and Salmassi ( [5]) proved the following result.
Using this, we get the following sufficient condition for uniform starlikeness Proof Let f be analytic in U and normalized by This is equivalent to Thus by Theorem 2.1, f ∈ UST . This result generates many examples of uniformly starlike functions.

Example 2.2 Let
This function plays an important role in the class UCV of uniformly convex functions. It is known (see [3,7]) that f ∈ UCV if and only if The image of the unit disk U under P 2 is bounded by the parabola Moreover Using standard methods of integration after some calculations we get that belongs to the class UST .

Example 2.3 Let
Thus the function is in the class UST .

Example 2.4 Let
This function maps the unit disk onto the domain bounded by the hyperbola and clearly P 4 (U ) ⊂ {w ∈ C : |Arg w| < π/4}. Thus is uniformly starlike in U , because of f 4 (z) = P 4 (z) for z ∈ U .

Example 2.5 Let
The image of the unit disk U under P 5 is bounded by the lemniscate and is contained in the region {w ∈ C : |Arg w| < π/4}. Hence the function Corollary 3.1 (Corollary 3.1d.1 in [6]) Let h be starlike in U , with h(0) = 0 and let p(z) = a + a n z n + a n+1 z n+1 + . . . be analytic in U with a = 0. If and q is the best (a, n)-dominant.

CV(ϕ 3 ) ⊂ UST ,
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