Abstract

In this paper, we determine the radius of -uniform convexity, -starlikeness, and -convexity of order for the Weierstrass canonical product of an entire function as a root having smallest modulus and argument of a functional equation. As special cases, we also determine the radius of -uniform convexity, -starlikeness, and -convexity of order for the entire function .

1. Introduction

Let be a real number and be the class of analytic functions defined in the disk and satisfy the normalization conditions Let , where be a sequence with

where means the radius of convergence of the series . If then

In 1999, Kanas and Wisniowska [9] (also refer Goodman [7, 8], Rønning [15], and Ma and Minda [12]) proposed the idea of -uniform convexity denoted by

A function is said to be in , the class of -uniformly Convex of order [3], iff

A function is said to be in , the class of -starlike function of order [10], iff

Geometrically, the conditions (2) and (3) mean that for and , the images of under the functions and are in the conic domain contained in the right half plane for which and is the curve defined by the equation

Moreover, is an elliptic region for , parabolic for , and hyperbolic for , and finally, is the whole right half plane.

The radius of -uniform convexity of order denoted by and radius of -starlikeness of order denoted by are defined by

where

By specializing the parameters, we observe , radius of convexity, , radius of convexity of order , , radius of starlikeness of order and , radius of starlikeness.

Let and . A function is said to be in , the class of -convex functions (Mocanu functions) of order [14, 16] iff

The radius of -convexity (Mocanu functions) of order denoted by is defined by, for ,

Addressing radius problems for some special functions is a new direction in the geometric function theory. For recent studies on radius problems, we refer to [2, 4, 6, 11].

By the Weierstrass factorization theorem [18], the function

is an entire function for a proper choice of with zeros and no other zeros, where is an entire function with , , are certain nonnegative integers, and for each in which , the value of exponential factor becomes 1.

The product (9) is called the canonical Weierstrass product [1]. In Theorem 3 of [13] Merkes et al. determined the radius of starlikeness of the canonical Weierstrass product , and as a special case, the authors determined the radius of starlikeness of

Later in [17], Szasz obtained the radius of convexity for .

Motivated by the results of Szász [17] and Merkes et al. [13], we determine the radius of -uniformly convexity, -starlikeness, and -convexity of order for the function given by (9). Consequently, we also determine the radius of -uniform convexity, -starlikeness, and -convexity of order for the function in this paper. In order to prove the main result, we require the following lemma.

Lemma 1 (see [17]). If and , then

2. Main Results

Theorem 2. Let be a sequence with for , , and let be an analytic function in with and , for . If the function defined by is decreasing with respect to and is of the form (9) with for , then the radius of -starlikeness of order of the function is , the absolute value of the root of the equation having the smallest modulus and argument .

Proof. By logarithmic differentiation, (9) becomes For and , Since , (12) along with (13) implies

Also,

From (14) and (15), we have

By the virtue of minimum principle for harmonic functions,

We observe that the function defined by

is strictly decreasing; also, and .

Hence, the equation has a unique root in , and this root is

Remark 3. in Theorem 2 means that, if , then the image of under the function is in conic domain contained in the right half plane for which and is the curve defined by equation (4).

In the following remarks, we deduce the radius of some special classes by specializing the parameters in Theorem 2.

Remark 4. Taking in Theorem 2, we get , the radius of -starlikeness of the function . is the absolute value of the root of the equation having the smallest modulus and argument .

Remark 5. Letting in Theorem 2, we get , the radius of starlikeness of order of the function . is the absolute value of the root of the equation having the smallest modulus and argument .

In the following, we obtain the radius of -uniform convexity of order for .

Theorem 6. Let be a sequence with for , , and let be an analytic function in with , , and for . If the function defined by is decreasing, the function defined by is increasing with respect to and is of the form (9) with for ; then, -uniform convexity of order of the function is , the absolute value of the root of the equation having the smallest modulus and argument .

Proof. From (9),

Using (14) and the inequality of Lemma 1, we have

From (14), (20), and (21),

Also, we have From (22) and (23),

By the virtue of minimum principle for harmonic functions, where The function , defined by is strictly decreasing; also, observe that , . Thus, it follows that the equation has a unique root situated in , and this root is .

Remark 7. As and we have , which means that if , then the image of under the function contained in the right half plane for which is in hyperbolic domain contained in the right half plane for which and is the curve defined by equation (4).

By specializing the parameters in Theorem 6, we have

Remark 8. Substituting and in Theorem 6, we get the radius of -uniform convexity given by the absolute value of the root of the equation having the smallest modulus and argument .

Remark 9 (see [17]). Taking in Theorem 6, we get the radius of convexity of order given by the absolute value of the root of the equation , having the smallest modulus and argument .

Theorem 10. Let be a sequence with for , and let be an analytic function in with , , and for . If the function defined by is decreasing, the function defined by is increasing with respect to , and is of the form (9) with for and ; then, the radius of -convexity of order of the function is the smallest positive root of the equation having the smallest modulus and argument .

Proof. Consider

for every , and the equality holds for . By the virtue of minimum principle for harmonic functions,

Also, is strictly decreasing; also, and . Hence, the equation has a unique root in , and this root is .

Remark 11 (see [17]). Taking in Theorem 10, we get the radius of starlikeness of order , given by the absolute value of the root of the equation , having the smallest modulus and argument .

Remark 12 (see [17]). Taking , in Theorem 6, we get the radius of convexity of order given by the absolute value of the root of the equation , having the smallest modulus and argument .

In the following remark, we discuss the radius of -starlikeness, -uniform convexity, and -convexity of order for the function .

Remark 13. Let where is the Euler-Mascheroni constant [5], and let ,, and . Then,

We now have , , and it is easy to verify with equality iff and , . The conditions of Theorem 2, Theorem 6, and Theorem 10 are satisfied.

By Theorem 2, the radius of -starlikeness of order of the function is the modulus of the biggest negative root of the equation Numerical approach gives , , , , and

By Theorem 6, the radius of -uniform convexity of order of the function is the modulus of the biggest negative root of the equation

Numerical approach gives , , , and

By Theorem 10, the radius of -convexity of order for the function is the modulus of the biggest negative root of the equation

Numerical approach gives , , , , and

In the following remark, we give an example which shows that the Theorems 2, 6, and 10 work even if is not starlike. That is, the example given in the following remark shows that the hypotheses of Theorems 2, 6, and 10 are free from the hypothesis of Theorem 3 in [13], proved by Merkes et al.

Remark 14. Let with , and let . Clearly, is not starlike. Then, we have and . Also, and , with equality iff .

By Theorem 2, the radius of -starlikeness of order of the function is the smallest positive root of the equation Numerical approach gives , , , , and

By Theorem 6, the radius of -uniform convexity of order is the smallest positive root of the equation

Numerical approach gives , , , and

By Theorem 10, the radius of -convexity of order is the smallest positive root of the equation

Numerical approach gives , , , , , , and

Data Availability

No data were used to support this study.

Conflicts of Interest

There is no conflict of interest regarding the publication of this article.