Radius Constants for Analytic Functions with Fixed Second Coefficient
Let be analytic in the unit disk with the second coefficient satisfying , . Sharp radius of Janowski starlikeness is obtained for functions whose th coefficient satisfies or . Other radius constants are also obtained for these functions, and connections with earlier results are made.
Let denote the class of analytic functions defined in the open unit disk , normalized by , and let denote its subclass consisting of univalent functions. If , de Branges  obtained the sharp coefficient bound that . However, the inequality , , is not sufficient for to be univalent; for example, is clearly not a member of .
Several subclasses of 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 [2–4]. 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 [5–7]. The th coefficient of close-to-convex functions with argument satisfies , and the coefficients of uniformly starlike functions are bounded by , while  for uniformly convex functions. Simple examples show that these bounds are not sufficient to characterize the geometric properties of the classes of functions.
In the sequel, we will assume that has the Taylor expansion of the form . Gavrilov  showed that the radius of univalence for functions satisfying is the real root of the equation , and the result is sharp for . Gavrilov also proved that the radius of univalence for functions satisfying the coefficient bound is . The condition clearly holds for functions satisfying , and for these functions, Landau  proved that the radius of univalence is . In fact, Yamashita  showed that the radius of univalence obtained by Gavrilov  is also the radius of starlikeness for functions satisfying or . Additionally, Yamashita  determined that the radius of convexity for functions satisfying is the real root of the equation , while the radius of convexity for functions satisfying is the real root of Recently, Kalaj et al.  obtained the radii of univalence, starlikeness, and convexity for harmonic mappings satisfying certain coefficient inequalities.
For two analytic functions and , the function is subordinate to , denoted by , if there is an analytic self-map of with satisfying . If is univalent, then is equivalent to and .
The class of Janowski starlike functions  consists of satisfying the subordination Certain well-known subclasses of starlike functions are special cases of for appropriate choices of the parameters and . For example, for , is the familiar class of starlike functions of order . Denote by the class . Janowski  obtained the sharp radius of convexity for .
This paper studies the class consisting of functions , , in the disk . The subclass of univalent functions in have been studied in [30–33]. In , Ravichandran obtained sharp radii of starlikeness and convexity of order for functions satisfying or , . The author also obtained the radius of uniform convexity and parabolic starlikeness for functions satisfying , .
This paper finds radius constants for functions satisfying either or . In the next section, sharp -radius and -radius are derived for these classes. Several known radius constants are shown to be special cases of the results obtained.
2. Radius Constants
A sufficient condition for functions to belong to the class is given in the following lemma.
Making use of this lemma, the sharp -radius is obtained for satisfying the coefficient inequality .
Theorem 2. Let , , and . The -radius for satisfying the coefficient inequality , , , is the real root in of the equation For , this number is also the -radius of . The results are sharp.
Proof. The number is the -radius for if and only if . Therefore, by Lemma 1, it is sufficient to verify the inequality
where is the real root in of (6). Using the known expansions
For , consider the function At the root in of (6), satisfies where This shows that is the sharp -radius for . For , (14) shows that the rational expression is positive, and therefore the equality holds. Thus, is the sharp -radius for when .
For , the function demonstrates sharpness of the result. The derivation is similar to the case and is omitted.
Theorem 3. Let and . The -radius of satisfying the coefficient inequality for and is the real root in of the equation For , this number is also the -radius of . The results are sharp.
Proof. By Lemma 1, is the -radius of functions when inequality (7) holds for the real root of (18) in . Using (8) and (9) together with
To verify sharpness for , consider the function At the root in of (18), satisfies Thus, is the sharp -radius for . For , the rational expression in (22) is positive, and therefore which shows that is the sharp -radius for . For , sharpness of the result is demonstrated by the function given by
Remark 4. The results obtained above yield the following special cases. (1)For , , , , and , Theorem 2 yields the radius of starlikeness obtained by Yamashita .(2)For , , and , Theorem 2 reduces to Theorem 2.1 in [33, page 3]. When , , and , Theorem 2 leads to Theorem 2.5 in [33, page 5].(3)For , Theorem 3 yields the radius of starlikeness of order for obtained by Ravichandran [33, Theorem 2.8].
The following result of Goel and Sohi  will be required in our investigation of the class of Janowski starlike functions.
Lemma 5 (see ). Let . If satisfies the inequality then .
The next result finds the sharp -radius for satisfying the coefficient inequality .
Theorem 6. Let . The -radius for satisfying the coefficient inequality , and , is the real root in of the equation This radius is sharp.
Proof. It is evident that is the -radius of if and only if . Hence, by Lemma 5, it suffices to show that
where is the root in of (26). From (8), (9), and (10), it follows that
The function given by (13) shows that the result is sharp. Indeed, at the point where is the root in of (26), the function satisfies Then, (26) yields or equivalently .
Theorem 7. Let . The -radius for satisfying the coefficient inequality , and , is the real root in of the equation This radius is sharp.
Proof. By Lemma 5, condition (27) assures that is the -radius of where is the real root of (31). Therefore, using (8) and (19) for yields The result is sharp for the function given by (21). Indeed, satisfies at the root in of (31). Evidently, the function satisfies (30), and hence the result is sharp.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The work presented here was supported in parts by an FRGS Grant 203/PMATHS/6711366 and a grant from the University of Delhi.
R. Nevanlinna, “Uber die konforme Abbildung Sterngebieten,” Oeversikt av Finska-Vetenskaps Societeten Forhandlingar A, vol. 63, no. 6, pp. 1–21, 1921.View at: Google Scholar
A. W. Goodman, Univalent Functions, vol. 1, Mariner Publishing, Tampa, Fla, USA, 1983.View at: MathSciNet
C. Lowener, “Untersuchungen uber die Verzerrung bei konformen Abbildungen des Einheitskreises ,” Leipzing Berichte, vol. 69, pp. 89–106, 1917.View at: Google Scholar
E. Landau, “Der Picard-Schottkysche Satz und die Blochsche Konstante,” Sitzungsberichte der Preussischen Akademie der Wissenschaften. Physikalisch-Mathematische Klasse, pp. 467–474, 1925.View at: Google Scholar
Z. W. Liu and M. S. Liu, “Properties and characteristics of certain subclass of analytic functions,” Journal of South China Normal University: Natural Science Edition, no. 3, pp. 11–14, 18, 2010.View at: Google Scholar