In this paper, several Turán-type inequalities for the more generalized Mittag-Leffler function are proved. In addition, we also gave affirmative answers to two open problems posed by Mehrez and Sitnik.

1. Introduction

The Mittag-Leffler function was first defined by Mittag-Leffler in 1903 [1]. In this paper, he defined the function by where is a classical gamma function. In 1905, Wiman [2] generalized as

In 1971, Prabhakar [3] introduced the function

Later, Dorrego and Cerutti [4] introduced the -Mittag-Leffler function where is the -Pochhammer symbol defined by . Recently, a generalization of the -Mittag-Leffler function was introduced and studied in [5].

The Mittag-Leffler function plays an important role in various branches of applied mathematics and engineering sciences, such as chemistry, biology, statistics, thermodynamics, mechanics, quantum physics, informatics, and signal processing. In 1930, the most known result in this field is an explicit formula for the resolvent of Riemann-Liouville fractional integral proved by E. Hille and J. Tamarkin. Based on these important formulas, many results are based still for solving fractional integral and differential equations. More properties and numerous applications of the Mittag-Leffler function to fractional calculus are collected, for instance, in References [6, 7]. In particular, we also refer to References [3, 810]. On the recent introduction of the Mittag-Leffler function and its generalizations, the reader may see [11, 12]. There are further related generalizations of the Mittag-Leffler function.

Recently, Mehrez and Sitnik ([13, 14]) obtained some Turán-type inequalities for the Mittag-Leffler function by considering monotonicity for the special ratio of sections for series of the Mittag-Leffler function. In the course of their research, they used a new method. We call it the Mehrez-Sitnik method (the reader can refer to [1517]). And then they applied this method to the Fox-Wright function and got a lot of interesting new results.

Turán-type inequalities which initiated a new field of research on inequalities for special functions were proved by Paul Turán, it states where , , and stands for the classical Legendre polynomial.

In this paper, we mainly consider a more general generalization where and It is clear that and . In the following, we mainly prove the monotonicity of ratios for sections of series of generalized Mittag-Leffler functions; the result is also closely connected with Turán-type inequalities.

2. Definition of the -Gamma Function and Lemmas

In 2007, Díaz and Pariguan [18] defined the -analogue of the gamma function for and as where . Similarly, we may define the -analogue of the digamma and polygamma functions as

It is well known that the -analogues of the digamma and polygamma functions satisfy the following recursive formula and series identities (see [18]):

For more properties of these functions, the reader may see Reference [19].

Lemma 1 (see [16]). Let and be real numbers such that and is increasing (decreasing); then, is increasing (decreasing).

Lemma 2 (see [20]). Let and be real numbers and let the power series and be convergent if . If and the sequence is (strictly) increasing (decreasing), then the function is also (strictly) increasing (decreasing) on .

3. Main Results

Our results read as follows.

Theorem 3. Let , , , , , , and . We define the function on by Then, the Turán-type inequality holds true.

Proof. Using the formulas we have where On the other hand, we have Taking into account the inequality [21], which holds for all , , and , and clearly, we have . This in turn implies that inequality (13) holds.

Corollary 4. Let , , , , , and . Then,

Proof. By taking and in Theorem 3, we easily obtain the above Turán-type inequality. The proof is complete.

Remark 5. It is worth noting that Mehrez and Sitnik posed an open problem 1 in [13]: “find the generalization of the inequality in the following inequality where , , , , and .
Clearly, Corollary 4 gives an affirmative answer to problem 1 in [13].

Theorem 6. Let , , , , , , and . We define the function on by Then, the function is increasing on . So the Turán-type inequality is valid for , , , , , and . The constant on the left-hand side of inequality is sharp.

Proof. By using the Cauchy product, we have We define the sequence by where the sequence is defined by In [21], the authors proved that the sequence is increasing for all . Thus, By using the inequality we have So the sequence is increasing for , and , , , ,, and . This implies that the ratios increase by Lemma 1. So the function is increasing on by Lemma 2. Finally, it is easy to see that So the constant is the best possible for which the inequality holds for all , , , , , and .

Corollary 7. For , , , , , and , the function is increasing on .

Proof. By putting and , we can get the above inequality. The proof is complete.

Remark 8. In [13], Mehrez and Sitnik also posed another open problem 2: “for , find the monotonicity of the function where , , , , , and ?
Here, Corollary 7 gives an affirmative answer to problem 2 in [13].

Theorem 9. For , , , , , , , and fixed , the function is strictly log-convex on . As a result, we have the following inequality:

Proof. Simple computation yields where we apply that the function is concave on . Therefore, we get that the function is strictly log-convex on . Using the fact that the sum of the log-convex functions is also log-convex, we obtain that the function is strictly log-convex on .
Due to inequality (33), we easily know That is, Using the definition of , we easily obtain The proof is complete.

Corollary 10. For , , , , , , , and fixed , we have

Proof. Since the function is strictly log-convex, we obtain that the function is strictly increasing on . Taking , we have This completes the proof.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interests.

Authors’ Contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.


The authors were supported by the Natural Foundation of Shandong Province (Grant No. ZR2018MF023) and by the Science Foundation of Binzhou University (Grant Nos. BZXYLG1903 and BZXYL1506).