Radius Constants for Analytic Functions with Fixed Second Coefficient

Let f(z) = z + ∑n=2 ∞ a n z n be analytic in the unit disk with the second coefficient a 2 satisfying |a 2 | = 2b, 0 ≤ b ≤ 1. Sharp radius of Janowski starlikeness is obtained for functions f whose nth coefficient satisfies |a n | ≤ cn + d  (c, d ≥ 0) or |a n | ≤ c/n  (c > 0  and  n ≥ 3). Other radius constants are also obtained for these functions, and connections with earlier results are made.

Several subclasses of S possess a similar coefficient bound. For instance, the th coefficients of starlike functions, convex functions in the direction of imaginary axis, and close-to-convex functions satisfy | | ≤ ( ≥ 2) [2][3][4]. Other examples include functions which are convex, starlike of order 1/2, and starlike with respect to symmetric points. The th coefficients of these functions satisfy | | ≤ 1 ( ≥ 2) [5][6][7]. The th coefficient of close-to-convex functions with argument satisfies | | ≤ 1 + ( − 1) cos [8], and the coefficients of uniformly starlike functions are bounded by 2/ [9], while | | ≤ 1/ [10] for uniformly convex functions. Simple examples show that these bounds are not sufficient to characterize the geometric properties of the classes of functions.
The class ST[ , ] of Janowski starlike functions [29] consists of ∈ A satisfying the subordination This paper studies the class A consisting of functions The subclass of univalent functions in A have been studied in [30][31][32][33]. In [33], Ravichandran obtained sharp radii of starlikeness and convexity of order for functions ∈ A satisfying | | ≤ or | | ≤ , ≥ 3. The author also obtained the radius of uniform convexity and parabolic starlikeness for functions ∈ A satisfying | | ≤ , ≥ 3.
This paper finds radius constants for functions In the next section, sharp L( , )radius and ST[ , ]-radius are derived for these classes. Several known radius constants are shown to be special cases of the results obtained.

Radius Constants
A sufficient condition for functions ∈ A to belong to the class L( , ) is given in the following lemma.
Making use of this lemma, the sharp L( , )-radius is obtained for ∈ A satisfying the coefficient inequality | | ≤ + .
For < 1, this number is also the L 0 ( , )-radius of ∈ A . The results are sharp.
demonstrates sharpness of the result. The derivation is similar to the case < 1 and is omitted.
For < 1, this number is also the L 0 ( , )-radius of ∈ A . The results are sharp.
Proof. By Lemma 1, 0 is the L( , )-radius of functions ∈ A when inequality (7) holds for the real root 0 of (18) in (0, 1). Using (8) and (9) together with To verify sharpness for < 1, consider the function 4 The Scientific World Journal At the root = 0 in (0, 1) of (18), 0 satisfies Re ( 2 0 ( ) 0 ( ) Thus, 0 is the sharp L( , )-radius for ∈ A . For < 1, the rational expression in (22) is positive, and therefore 2 0 ( ) 0 ( ) which shows that 0 is the sharp L 0 ( , )-radius for ∈ A . For > 1, sharpness of the result is demonstrated by the function 0 given by Remark 4. The results obtained above yield the following special cases.
The next result finds the sharp ST[ , ]-radius for ∈ A satisfying the coefficient inequality | | ≤ + . ∈ A satisfying the coefficient inequality | | ≤ + , ≥ 3 and , ≥ 0, is the real root in (0, 1) of the equation This radius is sharp.
. Hence, by Lemma 5, it suffices to show that where 0 is the root in (0, 1) of (26). From (8), (9), and (10), it follows that The Scientific World Journal 5 The function 0 given by (13) shows that the result is sharp. Indeed, at the point = 0 where 0 is the root in (0, 1) of (26), the function 0 satisfies Then, (26) yields This radius is sharp.
Proof. By Lemma 5, condition (27) assures that 0 is the ST[ , ]-radius of ∈ A where 0 is the real root of (31). Therefore, using (8) and (19) for ∈ A yields The result is sharp for the function 0 given by (21). Indeed, at the root = 0 in (0, 1) of (31). Evidently, the function 0 satisfies (30), and hence the result is sharp.