#### Abstract

For an analytic univalent function in the unit disk, it is well-known that for . But the inequality does not imply the univalence of . This motivated several authors to determine various radii constants associated with the analytic functions having prescribed coefficient bounds. In this paper, a survey of the related work is presented for analytic and harmonic mappings. In addition, we establish a coefficient inequality for sense-preserving harmonic functions to compute the bounds for the radius of univalence, radius of full starlikeness/convexity of order () for functions with prescribed coefficient bound on the analytic part.

#### 1. Introduction

Let denote the class of all analytic functions defined in the open unit disk normalized by . For functions of the form belonging to the subclass of consisting of univalent functions, de Branges [1] proved the famous Bieberbach conjecture that for . However, the inequality () does not imply that is univalent. A function given by (1) whose coefficients satisfy for is necessarily analytic in by the usual comparison test and hence a member of . But it need not be univalent. For example, the function satisfies the inequality () but its derivative vanishes inside and so the function is not univalent in . It is therefore of interest to determine the largest subdisk 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 and of , the -radius in is the largest such that, for every , for each .

##### 1.1. Analytic Case

Most of the classes in univalent function theory are characterized by the quantities or lying in a given domain in the right half-plane. For instance, the subclasses and () of consisting of starlike functions of order and convex functions of order , respectively, are defined analytically by the equivalences These classes were introduced by Robertson [2]. The classes and are the familiar classes of starlike and convex functions, respectively. Goodman [3] introduced the class of uniformly convex functions , which map every circular arc contained in with center onto a convex arc. For , Rønning [4] and Ma and Minda [5] independently proved that Closely related to the class is the class of parabolic starlike functions, introduced by Rønning [4] consisting of functions where ; that is, a function satisfies

In 1970, Gavrilov [6] showed that the radius of univalence for functions satisfying () is the real root of the equation . In 1982, Yamashita [7] showed that the radius of univalence obtained by Gavrilov [6] is also the radius of starlikeness for functions satisfying . Yamashita [7] also proved that the radius of convexity for functions satisfying () is the real root of the equation .

The inequality holds for functions satisfying . Gavrilov [6] proved that the radius of univalence for functions satisfying () is , which also turned out to be their radius of starlikeness, a result proved by Yamashita [7]. The radius of convexity for functions satisfying () is the real root of the equation .

For , let denote the class of functions given by (1) with . Since the second coefficient of normalized univalent functions determines their important properties such as Koebe-one-quarter theorem, growth and distortion theorems, the last author [8] obtained the sharp , (, and radii for functions satisfying , , or () for . Observe that a function with satisfies for . Indeed, Ravichandran [8] proved the following theorem, which includes the results of Gavrilov [6] and Yamashita [7] as special cases.

Theorem 1 (see [8]). *Let be given by (1) with for . Then we have the following.*(i)* satisfies the inequality
in where is the real root in of the equation . In particular, the number is also the radius of starlikeness of order and the number is the radius of parabolic starlikeness of the given functions.*(ii)* satisfies the inequality
in where is the real root in of the equation . In particular, the number is also the radius of convexity of order and the number is the radius of uniform convexity of the given functions.**The results are sharp for the function
*

It is observed that [9] if a function satisfies for , then . Similarly, Reade [10] proved that a close-to-star function satisfies for . However, the converse in both the cases is not true, in general. Recently, Mendiratta et al. [11] obtained sharp radii of starlikeness of order (), convexity of order (), parabolic starlikeness and uniform convexity for the class when or () for . Ali et al. [12] also worked in the similar direction and obtained similar radii constants.

##### 1.2. Harmonic Case

In a simply connected domain , a complex-valued harmonic function has the representation , where and are analytic in . We call the functions and the analytic and the coanalytic parts of , respectively. Let denote the class of all harmonic functions in normalized so that and take the form Since the Jacobian of is given by , by a theorem of Lewy [13], is sense-preserving if and only if , or equivalently if and the second dilatation satisfies in . Let be the subclass of consisting of sense-preserving functions. Then it is easy to see that for functions in the class . Set and . Finally, let and be subclasses of and , respectively, consisting of univalent functions.

One of the important questions in the study of class and its subclasses is related to coefficient bounds. In 1984, Clunie and Sheil-Small [14] conjectured that the Taylor coefficients of the series of and satisfy the inequality and it is still open. These researchers proposed this coefficient conjecture because the harmonic Koebe function where is expected to play the extremal role in the class . However, this conjecture is proved for all functions with real coefficients and all functions for which either is starlike with respect to the origin, close-to-convex, or convex in one direction (see [14–16]).

If for which is convex, Clunie and Sheil-Small [14] proved that the Taylor coefficients of and satisfy the inequalities and equality occurs for the harmonic half-plane mapping

Let and be subclasses of consisting of functions for which is convex and is starlike with respect to origin, respectively. Recall that convexity and starlikeness are not hereditary properties for univalent harmonic mappings (see [17–19]). 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 . Let and () be subclasses of consisting of fully starlike functions of order and fully convex functions of order , with and . The functions in the classes and are characterized by the conditions and for every circle , , respectively, where , .

The radius of full convexity of the class is while the radius of full convexity of the class is (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 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 . These results are generalized in context of fully starlike and fully convex functions of order () in [18]. The authors [18] proved the following result.

Theorem 2 (see [18]). *Let and have the form (9) with and . Then we have the following.*(a)*If the coefficients of the series satisfy the conditions (10), then is univalent and fully starlike of order in the disk , where is the real root in of the equation .*(b)*If the coefficients of the series satisfy the conditions (12), then is univalent and fully starlike of order in the disk , where is the real root in of the equation .**Moreover, the results are sharp for each .*

Theorem 2 gives the bounds for the radius of full starlikeness of order ( for the classes and . In addition, the authors in [18] also determined the bounds for the radius of full convexity of order ( for these classes.

The analytic part of harmonic mappings plays a vital role in shaping their geometric properties. For instance, if and is convex univalent, then and maps onto a close-to-convex domain (see [14, Theorem 5.17, p. 20]). However, if where and are given by (9) and for , then need not be even univalent; for example, the function belongs to but is not univalent in since where . Note that a convex univalent function satisfies for . This paper aims to determine the coefficient inequalities and radius constants for certain subfamilies of with the prescribed coefficient bound on the analytic part.

A coefficient inequality for functions in the class is obtained in Section 2 which, in particular, improves the coefficient inequality proved by Polatoğlu et al. [22] for perturbed harmonic mappings. Using this inequality, the bounds for the radius of univalence, full starlikeness, and full convexity of order () are obtained for functions where the coefficients of the analytic part satisfy one of the conditions , , or for . 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 ) are determined for a function , where 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.

#### 2. A Coefficient Inequality and Radius Constants

Firstly, we will obtain a coefficient inequality for functions in the class .

Theorem 3. *Let , where and are given by (9). Then
**
for , with . In particular, one has
*

*Proof. *Since , the function is analytic in and in . On equating the coefficients of in , we obtain
where . Since (see [23, p. 172]), it immediately follows that
Since , the desired result follows.

For specific choices of the analytic part in a harmonic function , Theorem 3 yields the following result.

Corollary 4. *Let , where and are given by (9). Then we have the following. *(i)*If or, in particular, is univalent, then , .*(ii)*If or, in particular, is convex univalent, then , .*(iii)*If or, in particular, , then , .*

*Remark 5. *Polatoğlu et al. [22] determined the coefficient inequality for harmonic functions in a subclass of , for which the analytic part is a univalent function in . They proved that if where and are given by (9) and if is univalent in , 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 () for the class 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 and () that directly follow from the corresponding results in [24, 25].

Lemma 6 (see [24, 25]). *Let , where and are given by (9) and let . Then we have the following.*(i)*If
then .*(ii)*If
then .*

Theorem 7. *Let , where and are given by (9) with and . Then we have the following. *(i)*If or, in particular, is univalent, then is univalent and fully starlike of order in the disk where is the real root of the equation
in the interval .*(ii)*If or, in particular, is convex univalent, then is univalent and fully starlike of order in the disk where is the real root of the equation
in the interval .*(iii)*If or, in particular, , then is univalent and fully starlike of order in the disk where is the real root of the equation
in the interval .*

*Proof. *Since , 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 , let be defined by
We will show that . In view of Lemma 6(i), it suffices to show that the sum
is bounded above by 1 for for .

(i) Since , it is easy to deduce that by (26). Using these coefficient bounds in (28) and simplifying, we have
Thus if satisfy the inequality
By using the identities
the last inequality reduces to
or equivalently
This gives
Thus by Lemma 6(i), for where is the real root of (23) in . In particular, is univalent and fully starlike of order in .

(ii) If then (26) gives . These coefficient bounds lead to the following inequality for the sum (28):
Therefore it follows that if satisfy the inequality
Making use of identities (31) in the last inequality, we obtain
which simplifies to
Lemma 6(i) shows that for where is the real root of (24) in . In particular, is univalent and fully starlike of order in .

(iii) Using (26), it is easily seen that . Using the coefficient bounds for and in (28), it follows that
The sum if satisfy the inequality
Using (31) and the identity , the last inequality reduces to
which is equivalent to
By applying Lemma 6(i), for where is the real root of (25) in . In particular, is univalent and fully starlike of order in . This completes the proof of the theorem.

*Remark 8. *By (26), it follows that for all functions . The bound is sharp for the function . Since is univalent in , the coefficient inequality remains sharp in the subclass . Clunie and Sheil-Small [14] were the first to observe the sharp inequality for functions in the class .

*Remark 9. *Let , where and are given by (9). In the proof of part (i) of Theorem 7, we noticed that if then . The bound for coincides with conjectured bound for when proposed by Clunie and Sheil-Small [14].

The next theorem calculates the radius of full convexity of order () for the class under certain choices of the coefficient bound on the analytic part.

Theorem 10. *Let , where and are given by (9) with and . Then we have the following. *(a)*If or, in particular, is univalent, then is fully convex of order in the disk where is the real root of the equation
in the interval .*(b)*If or, in particular, is convex univalent, then is fully convex of order in the disk where is the real root of the equation
in the interval .*(c)*If or, in particular, , then is fully convex of order in the disk where is the real root of the equation
in the interval .*

*Proof. *Following the method of the proof of Theorem 7, it suffices to show that the function defined by (27) belongs to . Using the coefficient bounds and , we deduce that
According to Lemma 6(ii), we need to show that , or equivalently
Using (31) and the identity , the last inequality reduces to
Lemma 6(ii) shows that for where is the real root of (43) in . In particular, is fully convex of order in . This proves (a). The proof of (b) and (c) follows on similar lines.

The sharpness of the radii constants for the class obtained in Theorems 7 and 10 is still unresolved. However, these constants can be shown to be sharp for certain subclasses of as seen by the following theorem.

Theorem 11. *Let () and let be a family of harmonic functions where and , given by (9) with , satisfy and for . Furthermore, if , , and denote, respectively, the radii of univalence, full starlikeness of order (), and full convexity of order () in , then we have the following.*(1)*If and , then , , and where and are the real roots of (23) and (43), respectively, in .*(2)*If and , then , , and where and are the real roots of (24) and (44), respectively, in .*(3)*If and , then , , and where and are the real roots of (25) and (45), respectively, in .*

*Proof. *Note that the roots of (23) in are decreasing as functions of . Consequently, . A similar remark holds for (24), (25), and (43)–(45). This observation together with Theorems 7 and 10 gives , , and for in the respective three cases specified in the theorem. Therefore it is enough to show that these radii constants are best possible.

(1) For sharpness of the numbers , let be defined by
As has real coefficients, for , the Jacobian of takes the form
Since the function is not univalent in if . Also, since
it follows that is not fully starlike of order in if , where is the real root of (23) in .

For sharpness of the numbers , consider the function
and observe that
This shows that is not fully convex of order in if , where is the real root of (43) in .

(2) The Jacobian of the function defined by
vanishes at and
These two observations imply that the numbers are sharp, where is the real root of (24) in . For sharpness of the constants , observe that the function
satisfies
where is the real root of (44) in .

(3) The function defined by
satisfies and
where is the real root of (25) in . If is the real root of (45) in , then
where is defined by

Now, we will discuss a particular case under which the results of Theorems 7 and 10 can be further improved.

Theorem 12. *Let , where and are given by (9) with . Further, suppose that the dilatation for all . Then we have the following. *(i)*If or, in particular, is univalent, then is univalent and fully starlike in the disk where is the real root of the equation in the interval . Moreover, is fully convex in where in the real root of the equation in the interval .*(ii)*If or, in particular, is convex univalent, then is univalent and fully starlike in the disk where . Also, is fully convex in where in the real root of the equation in the interval .*(iii)*If or, in particular, , then is univalent and fully starlike in the disk where . And is fully convex in where .*

*Proof. *Setting and for in (18), we obtain so that
Let be defined by (27). For the proof of (i), note that since , it is easily seen that using (62). Using these coefficient bounds, we have
using the identities (31). Thus if satisfy the inequality or . By Lemma 6(i), it follows that for where is the real root of in . In particular, is univalent and fully starlike in . For full convexity, observe that
The sum if satisfy the inequality . Thus Lemma 6(ii) shows that for where is the real root of in . In particular, is fully convex in . This proves (i). The other two parts of the theorem are similar and hence their proofs are omitted.

*Remark 13. *Observe that () and (). Here , , , and are as defined in Theorems 7, 10, and 12.

*Remark 14. *If , where and are given by (9) with and the dilatation is given by (), then
by setting and for in (18). Radius constants may be obtained in this case by carrying out a similar calculation as in the proof of Theorem 12.

#### 3. Sufficient Coefficient Estimates for Full Starlikeness and Convexity

In this section, we determine sufficient coefficient inequalities for functions to be in the classes and . As an application, these results are applied to hypergeometric functions in Section 4.

Theorem 15. *Let , where and are given by (9). Suppose that . Then one has the following.*(a)*If
then is fully starlike of order .*(b) *If
then is fully starlike of order *