Radius Constants for Functions with the Prescribed Coefficient Bounds

and Applied Analysis 3 Clunie and Sheil-Small [14] conjectured that the Taylor coefficients of the series of h and g satisfy the inequality 󵄨󵄨󵄨an 󵄨󵄨󵄨 ≤ 1 6 (2n + 1) (n + 1) , 󵄨󵄨󵄨bn 󵄨󵄨󵄨 ≤ 1 6 (2n − 1) (n − 1) , ∀n ≥ 2 (10) and it is still open.These researchers proposed this coefficient conjecture because the harmonic Koebe functionK = H+G where H(z) = z − (1/2) z2 + (1/6) z3 (1 − z)3


Introduction
Let A denote the class of all analytic functions  defined in the open unit disk D := { ∈ C : || < 1} normalized by (0) = 0 =   (0) − 1.For functions  of the form belonging to the subclass S of A consisting of univalent functions, de Branges [1] proved the famous Bieberbach conjecture that |  | ≤  for  ≥ 2. However, the inequality |  | ≤  ( ≥ 2) does not imply that  is univalent.A function  given by (1) whose coefficients satisfy |  | ≤  for  ≥ 2 is necessarily analytic in D by the usual comparison test and hence a member of A. But it need not be univalent.For example, the function satisfies the inequality |  | ≤  ( ≥ 2) but its derivative vanishes inside D and so the function  is not univalent in D. It is therefore of interest to determine the largest subdisk || <  < 1 in which the functions  satisfying the inequality |  | ≤  are univalent.Motivated by this problem, various radii problems associated with analytic as well as harmonic functions having prescribed coefficient bounds have been studied and we present a brief review of the research on this topic.Recall that given two subsets F and G of A, the Gradius in F is the largest  such that, for every  ∈ F,  −1 () ∈ G for each  ≤ .
Theorem 1 (see [8]).Let  ∈ A  be given by (1) with |  | ≤  for  ≥ 3. Then we have the following.is also the radius of starlikeness of order  and the number  0 (1/2) is the radius of parabolic starlikeness of the given functions.
(ii)  satisfies the inequality            ()   ()          < 1 −  (7) in || <  0 where  0 =  0 () is the real root in (0, 1) In particular, the number  0 () is also the radius of convexity of order  and the number  0 (1/2) is the radius of uniform convexity of the given functions.

Harmonic Case.
In a simply connected domain Ω ⊂ C, a complex-valued harmonic function  has the representation  = ℎ + , where ℎ and  are analytic in Ω.We call the functions ℎ and  the analytic and the coanalytic parts of , respectively.Let H denote the class of all harmonic functions  = ℎ +  in D normalized so that ℎ and  take the form Since the Jacobian of  is given by is expected to play the extremal role in the class S 0  .However, this conjecture is proved for all functions  ∈ S 0  with real coefficients and all functions  ∈ S 0  for which either (D) is starlike with respect to the origin, close-to-convex, or convex in one direction (see [14][15][16]).
If  ∈ S 0  for which (D) is convex, Clunie and Sheil-Small [14] proved that the Taylor coefficients of ℎ and  satisfy the inequalities and equality occurs for the harmonic half-plane mapping Let K 0  and S * 0  be subclasses of S 0  consisting of functions  for which (D) is convex and (D) is starlike with respect to origin, respectively.Recall that convexity and starlikeness are not hereditary properties for univalent harmonic mappings (see [17][18][19]).Chuaqui et al. [19] introduced the notion of fully starlike and fully convex harmonic functions that do inherit the properties of starlikeness and convexity, respectively.The last two authors [18] generalized this concept to fully starlike functions of order  and fully convex harmonic functions of order  for 0 ≤  < 1.Let FS *  () and FK  () (0 ≤  < 1) be subclasses of S  consisting of fully starlike functions of order  and fully convex functions of order , with FS *  := FS *  (0) and FK  := FK  (0).The functions in the classes FS *  () and FK  () are characterized by the conditions (/) arg (  ) >  and (/)(arg{(/)(  )}) >  for every circle || = ,  =   , respectively, where 0 ≤  < 2, 0 <  < 1.
The radius of full convexity of the class K 0  is √ 2 − 1 while the radius of full convexity of the class S * 0  is 3 − √ 8 (see [14,16,20]).The corresponding problems for the radius of full starlikeness are still unsolved.However, Kalaj et al. [21] worked in this direction and determined the radius of univalence and full starlikeness of functions  = ℎ +  whose coefficients satisfy the conditions (10) and (12).This, in turn, provides a bound for the radius of full starlikeness for convex and starlike mappings in S 0  .These results are generalized in context of fully starlike and fully convex functions of order  (0 ≤  < 1) in [18].The authors [18] proved the following result.
(a) If the coefficients of the series satisfy the conditions (10), then  = ℎ +  is univalent and fully starlike of order  in the disk || <   , where (b) If the coefficients of the series satisfy the conditions (12), then  = ℎ +  is univalent and fully starlike of order  in the disk || <   , where   =   () is the real root in (0, 1) of the equation Moreover, the results are sharp for each  ∈ [0, 1).
Theorem 2 gives the bounds for the radius of full starlikeness of order  (0 ≤  < 1) for the classes S * 0  and K 0  .In addition, the authors in [18] also determined the bounds for the radius of full convexity of order  (0 <  < 1) for these classes.
The analytic part of harmonic mappings plays a vital role in shaping their geometric properties.For instance, if  = ℎ+ ∈ H sp and ℎ is convex univalent, then  ∈ S  and maps D onto a close-to-convex domain (see [14,Theorem 5.17,p. 20]).However, if  = ℎ +  ∈ H sp where ℎ and  are given by (9) and |  | ≤ 1 for  ≥ 2, then  need not be even univalent; for example, the function A coefficient inequality for functions in the class H sp is obtained in Section 2 which, in particular, improves the coefficient inequality proved by Polatoglu et al. [22] for perturbed harmonic mappings.Using this inequality, the bounds for the radius of univalence, full starlikeness, and full convexity of order  (0 ≤  < 1) are obtained for functions  = ℎ +  ∈ H 0 sp where the coefficients of the analytic part ℎ satisfy one of the conditions In addition, we will also discuss a case under which these bounds can be improved.
In the third section, sharp bounds on  (depending upon  and | 1 |) are determined for a function  = ℎ +  ∈ H, where ℎ and  are given by ( 9), satisfying either of the following two conditions: to be either fully starlike of order  or fully convex of order .
The obtained results are applied to hypergeometric functions in Section 4.

A Coefficient Inequality and Radius Constants
Firstly, we will obtain a coefficient inequality for functions in the class H sp .
For specific choices of the analytic part ℎ in a harmonic function  = ℎ +  ∈ H sp , Theorem 3 yields the following result.
Polatoglu et al. [22] determined the coefficient inequality for harmonic functions in a subclass of H sp , for which the analytic part is a univalent function in D. They proved that if  = ℎ +  ∈ H sp where ℎ and  are given by ( 9) and if ℎ is univalent in D, then It is evident that Corollary 4(i) improves this bound.Now, we will determine the radius of univalence, radius of full starlikeness/full convexity of order  (0 ≤  < 1) for the class H 0 sp with specific choices of the coefficient bound on the analytic part.We will make use of the following sufficient coefficient conditions for a harmonic function to be in the classes FS *  () and FK  () (0 ≤  < 1) that directly follow from the corresponding results in [24,25].
univalent and fully starlike of order  in the disk || <  1 where  1 =  1 () is the real root of the equation in the interval (0, 1).
(ii) If |  | ≤ 1 or, in particular, ℎ is convex univalent, then  is univalent and fully starlike of order  in the disk || <  2 where  2 =  2 () is the real root of the equation in the interval (0, 1).
Proof.Since  = ℎ +  ∈ H 0 sp , we obtain by applying Theorem 3. We will make use of ( 26) to obtain the coefficient bounds for   in three different cases specified in the theorem.For  ∈ (0, 1), let   : D → C be defined by We will show that   ∈ FS *  ().In view of Lemma 6(i), it suffices to show that the sum is bounded above by 1 for 0 ≤  <   for  = 1, 2, 3. (26).Using these coefficient bounds in (28) and simplifying, we have Thus  ≤ 1 if  satisfy the inequality By using the identities the last inequality reduces to or equivalently Thus by Lemma 6(i),   ∈ FS *  () for  ≤  1 where  1 is the real root of ( 23) in (0, 1).In particular,  is univalent and fully starlike of order  in || <  1 .
Proof.Following the method of the proof of Theorem According to Lemma 6(ii), we need to show that   ≤ 1, or equivalently Using (31) and the identity 5 , the last inequality reduces to Lemma 6(ii) shows that   ∈ FK  () for  ≤  1 where  1 is the real root of (43) in (0, 1).In particular,  is fully convex of order  in || <  1 .This proves (a).The proof of (b) and (c) follows on similar lines.
The sharpness of the radii constants for the class H 0 sp obtained in Theorems 7 and 10 is still unresolved.However, these constants can be shown to be sharp for certain subclasses of H 0 as seen by the following theorem.
(1) For sharpness of the numbers  1 (), let   : D → C be defined by As   has real coefficients, for  ∈ (0, 1), the Jacobian of   takes the form it follows that   is not fully starlike of order  in || <  if  >  1 , where  1 =  1 () is the real root of ( 23) in (0, 1).
For sharpness of the numbers  1 (), consider the function and observe that This shows that   is not fully convex of order  in || <  if  >  1 , where  1 =  1 () is the real root of (43) in (0, 1).
is the real root of (45) in (0, 1), then where   : D → C is defined by Now, we will discuss a particular case under which the results of Theorems 7 and 10 can be further improved.

Sufficient Coefficient Estimates for Full Starlikeness and Convexity
In this section, we determine sufficient coefficient inequalities for functions to be in the classes FS *  () and FK  ().As an application, these results are applied to hypergeometric functions in Section 4.
The results are sharp. Proof.
By Lemma 6(i), it follows that  is fully starlike of order 2(1 − )/(2 + | 1 | + ).The harmonic function satisfies the coefficient inequality (66).Further, for  =   , we have which shows that the bound for the order of full starlikeness is sharp.This proves (a).
For the proof of (b), observe that using (67).Since  0 ∈ (0, 1),  is fully starlike of order of the theorem.For the order of full convexity of , note that where  0 is as defined in the proof of part (a) of the theorem.By Lemma 6(ii),  is fully convex of order 2(1−)/(2+| 1 |+).
In this case, the harmonic function shows that the result is best possible.