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Abstract
In this paper, our main aim is to discuss some monotonic and logarithmic concavity properties of three-
parameter Mittag-Leffler function by using its Weierstrassian product representation and some earlier
results on power series. In addition, by using the relationships between Mittag-Leffler type functions
and some basic functions, we give some specific examples related to the monotonic and logarithmic
concavity properties of some trigonometric and hyperbolic functions.
Results & Lemmas (9)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Lemma 1.1
Lemma 1.1 (see [1]) If ቀ 1 𝛼𝛼, 𝛽𝛽ቁ∈𝑊𝑊𝑖𝑖 and 𝛾𝛾> 0, then the function 𝑥𝑥↦𝜙𝜙(𝛼𝛼, 𝛽𝛽, 𝛾𝛾, −𝑥𝑥2) has infinitely many zeros which are all real.…
Lemma 1.1 (see [1]) If ቀ 1 𝛼𝛼, 𝛽𝛽ቁ∈𝑊𝑊𝑖𝑖 and 𝛾𝛾> 0, then the function 𝑥𝑥↦𝜙𝜙(𝛼𝛼, 𝛽𝛽, 𝛾𝛾, −𝑥𝑥2) has infinitely many zeros which are all real. Denoting by 𝜆𝜆𝛼𝛼,𝛽𝛽,𝛾𝛾,𝑛𝑛 the 𝑛𝑛th positive zero of 𝑥𝑥↦ 𝜙𝜙(𝛼𝛼, 𝛽𝛽, 𝛾𝛾, −𝑥𝑥2) under the same conditions the Weierstrassian decomposition
Lemma 1.2
Lemma 1.2 (see [2]) Consider the power series 𝑓𝑓(𝑥𝑥) = ∑𝑛𝑛≥0 𝑎𝑎𝑛𝑛𝑥𝑥𝑛𝑛 and 𝑔𝑔(𝑥𝑥) = ∑𝑛𝑛≥0 𝑏𝑏𝑛𝑛𝑥𝑥𝑛𝑛, where 𝑎𝑎𝑛𝑛∈ℝ and 𝑏𝑏𝑛𝑛> 0 for all 𝑛𝑛∈…
Lemma 1.2 (see [2]) Consider the power series 𝑓𝑓(𝑥𝑥) = ∑𝑛𝑛≥0 𝑎𝑎𝑛𝑛𝑥𝑥𝑛𝑛 and 𝑔𝑔(𝑥𝑥) = ∑𝑛𝑛≥0 𝑏𝑏𝑛𝑛𝑥𝑥𝑛𝑛, where 𝑎𝑎𝑛𝑛∈ℝ and 𝑏𝑏𝑛𝑛> 0 for all 𝑛𝑛∈{0,1, … }, and suppose that both converge on (−𝑟𝑟, 𝑟𝑟), 𝑟𝑟> 0. If the sequence { 𝑎𝑎𝑛𝑛 𝑏𝑏𝑛𝑛}𝑛𝑛≥0 is increasing(decreasing), then the function 𝑥𝑥↦ቀ 𝑓𝑓(𝑥𝑥) 𝑔𝑔(𝑥𝑥)ቁ is also increasing(decreasing) on (0, 𝑟𝑟).
Theorem 2.1
Theorem 2.1 Let ቀ 1 𝛼𝛼, 𝛽𝛽ቁ∈𝑊𝑊𝑖𝑖, 𝛾𝛾> 0 and denote the 𝑛𝑛th positive zero of the three-parameter Mittag-Leffler function 𝜙𝜙(𝛼𝛼, 𝛽𝛽, 𝛾𝛾,…
Theorem 2.1 Let ቀ 1 𝛼𝛼, 𝛽𝛽ቁ∈𝑊𝑊𝑖𝑖, 𝛾𝛾> 0 and denote the 𝑛𝑛th positive zero of the three-parameter Mittag-Leffler function 𝜙𝜙(𝛼𝛼, 𝛽𝛽, 𝛾𝛾, −𝑥𝑥2) by 𝜆𝜆𝛼𝛼,𝛽𝛽,𝛾𝛾,𝑛𝑛. Further, consider the following sets:
Corollary 2.2
Corollary 2.2 Let ቀ 1 𝛼𝛼, 𝛽𝛽ቁ∈𝑊𝑊𝑖𝑖 and denote the 𝑛𝑛th positive zero of the two-parameter Mittag- Leffler function 𝜙𝜙(𝛼𝛼, 𝛽𝛽, −𝑥𝑥2) by…
Corollary 2.2 Let ቀ 1 𝛼𝛼, 𝛽𝛽ቁ∈𝑊𝑊𝑖𝑖 and denote the 𝑛𝑛th positive zero of the two-parameter Mittag- Leffler function 𝜙𝜙(𝛼𝛼, 𝛽𝛽, −𝑥𝑥2) by 𝜆𝜆𝛼𝛼,𝛽𝛽,𝑛𝑛. Further, consider the following sets:
Corollary 2.3
Corollary 2.3 Let 𝛼𝛼> 1 and denote the 𝑛𝑛th positive zero of the one-parameter Mittag-Leffler function 𝜙𝜙(𝛼𝛼, −𝑥𝑥2) by 𝜆𝜆𝛼𝛼,𝑛𝑛. Further,…
Corollary 2.3 Let 𝛼𝛼> 1 and denote the 𝑛𝑛th positive zero of the one-parameter Mittag-Leffler function 𝜙𝜙(𝛼𝛼, −𝑥𝑥2) by 𝜆𝜆𝛼𝛼,𝑛𝑛. Further, consider the following sets:
Theorem 2.4
Theorem 2.4 Suppose that 𝛼𝛼> 1, 𝛽𝛽> 0 and 𝛾𝛾> 0. Let 𝜆𝜆𝛼𝛼,𝛽𝛽,𝛾𝛾,𝑛𝑛 denote the 𝑛𝑛th positive zero of the function 𝑥𝑥↦𝜙𝜙(𝛼𝛼, 𝛽𝛽, 𝛾𝛾,…
Theorem 2.4 Suppose that 𝛼𝛼> 1, 𝛽𝛽> 0 and 𝛾𝛾> 0. Let 𝜆𝜆𝛼𝛼,𝛽𝛽,𝛾𝛾,𝑛𝑛 denote the 𝑛𝑛th positive zero of the function 𝑥𝑥↦𝜙𝜙(𝛼𝛼, 𝛽𝛽, 𝛾𝛾, −𝑥𝑥2).Then, the function 𝑥𝑥↦𝜙𝜙(𝛼𝛼, 𝛽𝛽, 𝛾𝛾, −𝑥𝑥2) is strictly logarithmic concave on 𝐴𝐴3.
Corollary 2.5
Corollary 2.5 Suppose that 𝛼𝛼> 1 and 𝛽𝛽> 0. Then, the function 𝜙𝜙(𝛼𝛼, 𝛽𝛽, −𝑥𝑥2) is strictly logarithmic concave in 𝐵𝐵3, while the function…
Corollary 2.5 Suppose that 𝛼𝛼> 1 and 𝛽𝛽> 0. Then, the function 𝜙𝜙(𝛼𝛼, 𝛽𝛽, −𝑥𝑥2) is strictly logarithmic concave in 𝐵𝐵3, while the function 𝑥𝑥↦𝜙𝜙(𝛼𝛼, −𝑥𝑥2) is strictly logarithmic concave in 𝐶𝐶3. 166
Theorem 2.6
Theorem 2.6 Let suppose that 𝛼𝛼, 𝛽𝛽 and 𝛾𝛾 are positive real numbers. Then, the function
Theorem 2.6 Let suppose that 𝛼𝛼, 𝛽𝛽 and 𝛾𝛾 are positive real numbers. Then, the function
Theorem 2.4
Repeated OCR label — verify against source
Theorem 2.4 and Theorem 2.6. Some of these examples are given below:
Theorem 2.4 and Theorem 2.6. Some of these examples are given below:
Definitions (1)
Def 1.3
Definition 1.3 (see[6]) A function 𝑓𝑓 is said to be log-concave on interval (𝑎𝑎, 𝑏𝑏) if the function log 𝑓𝑓 is a concave function on (𝑎𝑎,…
Definition 1.3 (see[6]) A function 𝑓𝑓 is said to be log-concave on interval (𝑎𝑎, 𝑏𝑏) if the function log 𝑓𝑓 is a concave function on (𝑎𝑎, 𝑏𝑏).
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