Research Article | Open Access
Coefficient Conditions for Starlikeness of Nonnegative Order
Sufficient conditions on a sequence of nonnegative numbers are obtained that ensures is starlike of nonnegative order in the unit disk. A result of Vietoris on trigonometric sums is extended in this pursuit. Conditions for close to convexity and convexity in the direction of the imaginary axis are also established. These results are applied to investigate the starlikeness of functions involving the Gaussian hypergeometric functions.
Let denote the class of analytic functions defined in the unit disk normalized by the conditions . Denote by the subclass of consisting of functions univalent in . A function is starlike if is starlike with respect to the origin and convex if is a convex domain. These classes denoted by and , respectively, are subsets of . The generalized classes and of starlike and convex functions of order are defined, respectively, by the analytic characterizations with and .
An extension of starlike functions is the class of close-to-convex functions of order defined analytically by for some real . The family of close-to-convex functions of order with respect to is denoted by , with . Exposition on the geometric properties of functions in these classes can be found in [1, 2].
A function satisfying in is said to be typically real, and is convex in the direction of the imaginary axis if every line parallel to the imaginary axis either intersects in an interval or has an empty intersection. For with real coefficients, Robertson  proved that being convex in the direction of the imaginary axis is equivalent to being typically real, which in turn is equivalent to . For satisfying is typically real and in , Ruscheweyh  proved that it is necessarily starlike. The latter result is extended in  to include starlike functions of a nonnegative order.
Lemma 1.1 (see ). For , let satisfy and be typically real in . If in , then .
Trigonometric series, in particular the cosine and sine series along with their partial sums, have found widely important applications in many works, for example, those of [4–9]. Vietoris  (also see ) showed that if , , then for any positive integer . Here the Pochhammer symbol is defined by , and . Using Abel’s partial summation formula equation (1.3) yields the following classical result on the positivity of cosine and sine sums.
Theorem 1.2 (see ). Let be a decreasing sequence of nonnegative real numbers satisfying and . Then for any positive integer .
Using Theorem 1.2, Ruscheweyh  obtained sufficient coefficient conditions for functions to be starlike which can readily be tested. This paper aims to extend Ruscheweyh's concise result. Specifically in the next section, sufficient conditions on a sequence of nonnegative numbers are obtained that ensures is starlike of order in the unit disk. Coefficient conditions for to be either close to convex or convex in the direction of the imaginary axis are also derived. The final section is devoted to finding conditions on the triplets that will ensure a normalized Gaussian hypergeometric function is starlike of order , .
The following extension of Theorem 1.2 will be required.
Theorem 1.3 (see ). Let . For any positive integer and , then (i) if and only if ,(ii) if and only if ,(iii) if .
Here is the unique root in of
2. Main Results
For our purpose, it will be more expedient to allow the terms of the sequence in Theorem 1.3 to consist of nonnegative numbers. Thus as a prelude to the main results, Theorem 1.3 is first appropriately adapted to yield the following two preliminary results.
Lemma 2.1. Let be a decreasing sequence of nonnegative numbers satisfying and , . For any positive integer and , then
Proof. Let the sequence be given by . It is evident from Theorem 1.3(i) that
Using (1.4), rewrite in the form
If for , then a computation gives Similarly, Thus for . Together with (2.2), the latter implies that the expression on the right side of (2.3) is positive.
Now suppose there is an , , so that while for . Then the conditions on would imply that . If , evidently . Let . It is shown above in (2.6) that for . The conditions and imply that which yields the desired result.
The following result is readily obtained by using a similar argument used in Lemma 2.1.
Lemma 2.2. Let be a decreasing sequence satisfying and . If , then for any positive integer and .
The preceding lemmas will next be used to establish the following result on starlikeness.
Theorem 2.3. Let , satisfy . Then is starlike of order . The result is sharp as illustrated by the function .
Proof. Let , and . Then
With in (2.11), it follows that
Now and . Condition (2.9) shows that while inequality (2.10) yields Evidently satisfies the hypothesis of Lemma 2.1, and therefore The minimum principle for harmonic functions implies that is either identically zero or positive. Since at , it follows that in .
Similarly, taking in (2.11) results in for . Now Lemma 2.2 implies that Since the coefficients are real, (2.17) shows that on . Again by the minimum principle, is either identically zero or positive in . The former implies that , which is starlike. In the latter case, the reflection principle yields in . Thus is typically real.
It remains to show that is typically real satisfying . Let , and thus Inequality (2.9) yields while (2.10) implies Thus also satisfies the hypothesis of Lemmas 2.1 and 2.2, and following the same arguments used earlier, is deduced to be typically real with .
Lemma 1.1 now implies that is starlike of order . Since the class of starlike functions of a fixed order is a compact family, it is evident that is also starlike of order .
Finally note that when , then Hence the order of starlikeness is sharp.
For , Theorem 2.3 reduces to the following result of Ruscheweyh.
Corollary 2.4 (see ). Let satisfy Then is starlike.
Using Lemma 2.1 and the minimum principle for harmonic functions, the following sufficient condition for to be close to convex of order with respect to the starlike function is obtained.
Theorem 2.5. Let satisfy Then satisfies .
Proof. Let , and . Then
Letting , , in (2.25), it follows that
Employing the same argument used in the proof of Theorem 2.3, it is sufficient to consider only the interval .
Now implies , and inequality (2.23) shows that Also, since .
Inequality (2.24) also yields Thus satisfies the hypothesis of Lemma 2.1. The minimum principle for harmonic functions yields .
The next result gives a sufficient condition for to be convex in the direction of the imaginary axis, which is equivalent to with .
Theorem 2.6. Let satisfy . Then is convex in the direction of the imaginary axis whenever .
Proof. Since the coefficients of are real, is convex in the direction of the imaginary axis if and only if is typically real. Let . Then Inequality (2.29) shows that the coefficients satisfy the hypothesis of Lemma 2.2. Hence by taking , , in (2.30) and using Lemma 2.2, it follows that for . A similar argument used in the proof of Theorem 2.3 now leads to the conclusion that is typically real for .
Corollary 2.7. Let satisfy . Then is convex in the direction of the imaginary axis.
Proof. It is evident from Theorem 2.6 that is convex in the direction of the imaginary axis for any positive integer . The result now follows in light of the compactness of the class of functions convex in the direction of the imaginary axis.
Corollary 2.8 (see [6, Theorem , page 33]). Let , satisfy Then is convex in the direction of the imaginary axis.
3. Starlikeness of the Gaussian Hypergeometric Functions
For complex numbers , and with , the Gaussian hypergeometric function is defined by the series When or reduces to a hypergeometric polynomial of degree . Properties on the hypergeometric functions are treated in . The geometry of close to convexity, starlikeness, and convexity of has been studied in various works, for example, those of [6, 14–19]. Notwithstanding these works, the exact range of the triplets for starlikeness as well as for the other geometric structures of normalized Gaussian hypergeometric functions remains a formidable challenge.
In this section, conditions on the triplets are determined that will ensure the function is starlike of a certain order in . Several examples are presented to compare the range obtained with some of those earlier works. Sufficient conditions for starlikeness of the odd Gaussian hypergeometric functions are also obtained.
Theorem 3.1. Let and satisfy for . If , then is starlike of order .
Proof. The function can be expressed as
The sequence is first shown to satisfy conditions (2.9) and (2.10) in Theorem 2.3. Now consider with The conditions and show that , and . Thus and inequality (2.9) holds.
To verify inequality (2.10), consider with Again from the conditions and , computations show that Hence inequality (2.10) also holds. The desired result now readily follows from Theorem 2.3.
Choosing and , respectively, in Theorem 3.1 yields the following result.
Corollary 3.2. Let and be a fixed integer. (i)If , then is starlike of order . In particular, is starlike.(ii)If , then is starlike of order .
Observe that the starlikeness of follows from (i) by taking , , and .
For , Theorem 3.1 leads to the following result.
Corollary 3.3. Let satisfy for . If , then is starlike.
Remark 3.4. Sufficient conditions for starlikeness (of order 0) for are also obtained in [6, Theorem , page 60]. The result in Theorem 3.1 however investigated conditions for to be starlike of a certain order.
Corollary 3.5. Let and satisfy . If , then the function is convex of order .
Proof. If is starlike of a certain order, then Alexander’s transformation yields a function convex of the same order. Under the given hypothesis and using Theorem 3.1, it is clear that is starlike of order . The Gaussian hypergeometric function satisfies the identity Thus is convex of order .
We state the following recent result by Hästö et al.  related to the starlikeness of .
Theorem 3.6 (see ). Let , , and be nonzero real numbers such that has no zero in . Then is starlike of order , if (1), (2), (3),
where , , , , , and .
Next we provide some examples to show that Theorem 3.1 gives a better range of triplets than those obtained in earlier works.
Example 3.8. Let and . Then belongs to , which follows from Theorem 3.1 with and . Comparing with Theorem 3.6, note that and so the second condition in Theorem 3.6 does not hold. Therefore, the range of the parameters in Theorem 3.6 does not include the range in this example.
The next result gives conditions on triplets for which the odd Gaussian hypergeometric functions are starlike of order .
Theorem 3.9. Let satisfy . If , then is starlike of order .
Proof. Let and . Then Theorem 3.1 shows that , and therefore that is, .
For , Theorem 3.9 reduces to the following result.
Corollary 3.10. Let satisfy . Then is in for .
The work presented here was supported in part by a Research University Grant from Universiti Sains Malaysia.
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