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Certain geometric properties of Mittag-Leffler functions
Journal of Inequalities and Applications volume 2019, Article number: 94 (2019)
Abstract
In this paper, some geometric properties of normalized Mittag-Leffler functions are investigated. We focus on starlikeness of order \(2\mu +\eta -1\) and convexity in the direction of imaginary axis. In addition, we study pre-starlikeness of Mittag-Leffler functions. The results are obtained by using the positivity technique.
1 Introduction
Recently, there has been overwhelming interest in the study of Mittag-Leffler functions. The Mittag-Leffler type functions have widespread applications in physics, biology, chemistry, engineering, and some other applied sciences. Some other aspects of the applications of these functions can be seen in fractional differential equations, stochastic systems, dynamical systems, statistical distributions, and chaotic systems. The geometric properties like starlikeness, convexity, and close-to-convexity of these functions have been investigated on a large scale by a number of researchers. We can easily see the direct usage of these functions to many techniques of fractional calculus. Gorenflo et al. [9], Kilbas et al. [14], and Srivastava et al. [34, 35] are a few leading precedents of these contributions. The Swedish mathematician G. M. Mittag-Leffler gave the idea of so-called Mittag-Leffler function \(E_{\alpha }(z)\), see [17]. Later, it was studied by Wiman [37, 38] who defined it in terms of power series depending on the complex parameter α. It may be regarded as a special function of \(z\in \mathbb{C} \). We define it as follows:
We observe that the power series (1.1) converges in the whole complex plane for all \(\operatorname{Re}\alpha >0\), whereas it diverges everywhere on \(\mathbb{C} \backslash \{0\}\) for all \(\operatorname{Re}\alpha <0\). We also note that for \(\operatorname{Re}\alpha =0\), its radius of convergence is \(R=e^{\pi /2| \operatorname{Im}(z)|}\). Wiman [37, 38] gave the first two-parametric generalizations of the function defined in (1.1). It was later studied by Agarwal [4], Humbert [10], and Agarwal and Humbert [11]. We define it as follows:
For different values of parameters, the conditions of convergence vary. When α and β are positive and real, the above given series converges in the entire complex plane. Another important and interesting fact about Mittag-Leffler functions is that the one and two-parametric Mittag-Leffler functions are fractional extensions of the basic functions. That is, \(E_{1}(\pm z)=E_{1,1}(\pm z)=e^{\pm z}\), \(E_{1,2}(z)=(e^{z}-1)/z\), \(E_{2}(z)=E_{2,1}(z)=\cosh (\sqrt{z})\), \(E_{2,2}(z)=\sinh (\sqrt{z})/\sqrt{z}\). For more about Mittag-Leffler functions, see [1, 3, 12].
Let \(\mathcal{A}\) denote the class of functions f of the form
which are analytic in the open unit disc \(\mathcal{U}= \{ z \in \mathbb{C} : \vert z \vert <1 \} \). Let \(\mathcal{S}\) denote the class of all functions in \(\mathcal{A}\) which are univalent in \(\mathcal{U}\). Let \(\mathcal{S}^{\ast }(\mu )\), \(\mathcal{C}(\mu )\), and \(\mathcal{K}(\mu )\) denote the classes of starlike, convex, and close-to-convex of order μ, respectively, defined as follows:
and
It is clear that
The functions z, \(\frac{z}{ ( 1-z ) }\), \(\frac{z}{1-z^{2}}\), \(\frac{z}{1+z}\), \(\frac{z}{1+z^{2}}\) are starlike and univalent functions. It is convenient for f to be close-to-convex, when the corresponding function g has one of the aforementioned forms.
Consider a function \(f\in \mathcal{A}\) which is real on the segment \((-1,1)\). If f satisfies the relation
then f is called a typically real function. The class of typically real functions \(\mathcal{T}\) was introduced by Robertson [26]. A function \(f\in \mathcal{S}\) is said to be convex in the direction of imaginary axis if and only if the domain \(f ( \mathcal{U} ) \) is convex in the direction of imaginary axis. That is, for every \(w_{1},w_{2}\in f ( \mathcal{U} ) \), \([w_{1},w_{2}] \subset f ( \mathcal{U} ) \) such that \(\operatorname{Re}w_{1}=\operatorname{Re}w_{2}\). Robertson [26] showed that a function \(f\in \mathcal{A}\) with real coefficients is convex in the direction of imaginary axis if \(zf^{\prime } ( z ) \) is typically real. It is equivalent to
Ruscheweyh [27] proved that if a function f \(\in \mathcal{T}\) and satisfies \(\operatorname{Re}f^{\prime }(z)>0\) for \(z\in \mathcal{U}\), then f is a starlike univalent function in \(\mathcal{U}\). The extension of this definition up to order μ was given by Mondal and Swaminathan in [18].
Let \(f\in \mathcal{A}\) of the form (1.3) and \(g\in \mathcal{A}\) be given by
Then convolution or Hadamard product of f and g is defined as
We also focus on the class of pre-starlike functions, initiated in [28]. The class \(\mathcal{R}_{\mu }\) denotes the class of pre-starlike functions of order μ and is defined as follows:
In particular, \(\mathcal{R}_{1/2}=\mathcal{S}^{\ast } ( 1/2 ) \) and \(\mathcal{R}_{0}=\mathcal{C}\). Sheil-Small et al. [32] generalized the class \(\mathcal{R}_{\mu }\) and defined the class \(\mathcal{R}[\rho ,\mu ]\). A function \(f\in \mathcal{A}\) is in the class \(\mathcal{R}[\rho ,\mu ]\) if \(f\ast \mathcal{S}_{\rho }\in \) \(\mathcal{S}^{\ast }(\mu )\), where \(\mathcal{S}_{\rho }=\frac{z}{(1-z)^{2-2 \rho }}\), \(0\leq \rho <1\). It is easy to see that \(\mathcal{R}[\mu , \mu ] = \mathcal{R}_{\mu }\). For more details, see [8, 29].
Observe that Mittag-Leffler function \(\mathbb{E}_{\alpha ,\beta }\) does not belong to the family \(\mathcal{A}\). Thus, it is natural to consider the following normalization of Mittag-Leffler functions:
Formula (1.4) holds for complex parameters α, β, and \(z\in \mathbb{C} \). In this paper, we shall restrict our attention to the case of real-valued α, β, and \(z\in \mathcal{U}\). For particular values of α and β, we obtain several functions, for example
In some recent years, several researchers studied geometric properties such as starlikeness, convexity, and close-to-convexity of certain special functions; for details, see [5,6,7, 19, 21,22,23,24,25, 36] and the references therein. Also see [2, 13, 16, 30, 33] for some properties of special functions and mathematical inequalities. More recently, Sangal and Swaminathan [31] studied geometric properties of hypergeometric functions by using the positivity technique.
In this paper, we study pre-starlikeness and deduce the convexity and starlikeness of order \(1/2\). We also investigate the starlikeness of order \(2\mu +\eta -1\). Furthermore, we find the convexity of Mittag-Leffler functions in the direction of imaginary axis. The main tool in this investigation is the positivity technique.
2 Preliminaries
To obtain our main results, we need the following lemmas.
Lemma 2.1
([18])
Let \(\{ a_{k} \} _{k=1}^{\infty }\) be a sequence of positive numbers such that \(a_{1}=1\). If, for \(0\leq \mu <1\),
-
(1)
\(( 1-\mu ) a_{1}\geq ( 2-\mu ) a_{2} \geq 2^{ ( \mu +1 ) } ( 3-\mu ) a_{3}\),
-
(2)
\(( k-1-\mu ) ( k-\mu ) a_{k}\geq k ( k+1-\mu ) a_{k+1}\), \(\forall k\geq 3\).
Then \(f ( z ) =z+\sum^{\infty }_{k=2} a _{k}z^{k}\in \mathcal{S}^{\ast } ( \mu ) \).
Lemma 2.2
([20])
Let \(\eta \geq 0\), \(\mu \in \mathbb{R} \) such that \(0<\mu +\eta <1\) and \(n\in \mathbb{N} \). If \(d_{0}=d_{1}=1\) and \(d_{2k}=d_{2k+1}=\frac{(1+\eta )_{n-k}n!}{(n-k)!(1+\eta )_{n}}.\frac{( \mu +\eta )_{k}}{k!}\) for \(1\leq k\leq n\), then
-
(i)
\(\sum_{k=0}^{n}d_{k}\cos (k\theta )>0\Leftrightarrow \mu +\eta \leq \mu ^{\ast } ( \frac{1}{2} ) =0.691556\ldots\) ,
-
(ii)
\(\sum_{k=1}^{2n+1}\sin (k\theta )>0\Leftrightarrow \mu +\eta \leq \mu ^{\ast } ( \frac{1}{2} ) \),
-
(iii)
\(\sum_{k=1}^{2n}\sin (k\theta )>0\) for \(\mu +\eta \leq \frac{1+\eta }{2}\),
where \(\mu ^{\ast }(\gamma )\), \(\gamma \in ( 0,1 ] \) is the unique solution in \(] 0,1 [ \) of \(\int _{0}^{ ( \gamma +1 ) \pi }\frac{\sin ( t-\gamma \pi ) }{t ^{1-\mu }}\,dt=0\).
Koumandos and Ruscheweyh [15] obtained the value of \(\mu ^{ \ast }(\gamma )\). In this work, we use the particular value \(\mu ^{\ast } ( \frac{1}{2} ) =\mu _{0}^{\ast }\).
Lemma 2.3
([31])
Let \(0\leq \eta \leq 2\mu _{0}^{\ast }-1\), \(\mu \in \mathbb{R} \) such that \(0<\mu +\eta <1\) and \(n\in \mathbb{N} \). If \(\{ a_{k} \} _{k=1}^{\infty }\) is a decreasing sequence of non-negative numbers satisfying \(a_{0}>0\) and
then, for all \(0<\theta <\pi \),
Lemma 2.4
([31])
Let \(0\leq \eta \leq 2\mu _{0}^{\ast }-1\) and \(-\eta < \mu \leq \frac{1-\eta }{2}\), \(a_{1}=1\), \(a_{k}\geq 0\) satisfy
for \(1\leq k\leq n\). Then \(f_{n}(z)=\sum_{k=1}^{n}a_{k}z^{k}\) is starlike of order \(\frac{1-2\mu -\eta }{(1+\eta )(1-\mu -\eta )}\). Moreover, in the limiting case, \(f(z)=\lim_{n\rightarrow \infty }f _{n}(z)=\sum_{k=1}^{\infty }a_{k}z^{k}\) is starlike of the same order if \(\{ a_{k} \} \) satisfy (2.1) and in addition
Lemma 2.5
([31])
Let \(\mu \in \mathbb{R} \) and \(\eta \geq 0\) such that \(0<\mu +\eta <1\), and let \(a_{1}=1\) and \(a_{k}\geq 0\) satisfy
and
Then \(f_{n}(z)=z+\sum_{k=2}^{n}a_{k}z^{k}\) satisfies \(\operatorname{Re}(f_{n}^{\prime }(z))>1-\frac{\mu +\eta }{\mu _{0}^{\ast }}\).
3 Main results
Theorem 3.1
Let \(\alpha \geq 1\), \(\beta \geq 1\). Then \(E_{\alpha ,\beta } ( z ) \in \mathcal{R} [ \rho ,\mu ] \) for \(0\leq \mu <1\) and
where
Proof
Consider the function \(g ( z ) =z+\sum^{\infty }_{k=2} b_{k}z^{k}\), where \(b_{k}\) is given by
Now
implies \(\varGamma ( \alpha +\beta ) \geq T_{1} ( \rho ,\mu ) \varGamma ( \beta ) \), where \(T_{1} ( \rho ,\mu ) =\frac{2 ( 2-\mu ) ( 1- \rho ) }{ ( 1-\mu ) }\). Again
with \(\frac{\varGamma ( \alpha +\beta ) }{\varGamma ( \beta ) }\geq 2T_{2} ( \rho ,\mu ) \frac{\varGamma ^{2} ( \alpha +\beta ) }{\varGamma ( \beta ) \varGamma ( 2\alpha +\beta ) }\), where \(T_{2} ( \rho ,\mu ) =\frac{2^{\mu -1} ( 3-\mu ) ( 3-2 \alpha ) }{ ( 2-\mu ) }\). Also consider
where \(A ( k ) =\frac{b_{k}}{\varGamma ( \alpha k+ \beta ) }\) and
Here
This implies that
where \(T_{3} ( \rho ,\mu ) =\frac{2 ( 4-\mu ) ( 2-\alpha ) }{ ( 2-\mu ) ( 3-\mu ) }\). It is clear that \(A ( \rho ,\mu )\), \(B ( \rho ,\mu )\), \(D ( \rho ,\mu )\) are non-negative. Since each coefficient of \(( k-3 ) \) and the constant term in \(M ( k ) \) are non-negative, therefore \(M ( k ) \) is an increasing function for \(k\geq 3\). Also, for \(M ( 3 ) >0\), we have \(( k-1-\mu ) ( k- \mu ) b_{k}\geq k ( k+1-\mu ) b_{k+1}\). Thus \(b_{k}\) satisfies the conditions of Lemma 2.1 and hence \(g\in \mathcal{S}^{\ast } ( \mu ) \). After simple computations, we observe that \(g ( z ) =E_{\alpha ,\beta } ( z ) \ast \frac{z}{ ( 1-z ) ^{2-2\rho }}\). Therefore, by the definition of \(\mathcal{R} [ \rho ,\mu ]\), we have \(E_{\alpha ,\beta } ( z ) \in \mathcal{R} [ \rho , \mu ] \). Now consider
The numerator is negative for all μ and hence \(T_{3} ( \rho ,\mu ) \leq T_{1} ( \rho ,\mu ) \) for \(0\leq \rho \leq \rho _{0} ( \mu ) \). Similarly, if \(0\leq \rho \leq \rho _{1} ( \mu ) \), \(T_{1} ( \rho , \mu ) \geq T_{2} ( \rho ,\mu ) \) for all μ. Here,
Clearly, we can investigate that, for \(0\leq \rho \leq \min \{ \rho _{0} ( \mu ) ,\rho _{1} ( \mu ) \} \),
Now, we only need to check the \(\min \{ \rho _{0} ( \mu ) ,\rho _{1} ( \mu ) \} \). Consider
where
This implies \(\rho _{0} ( \mu ) =\min \{ \rho _{0} ( \mu ) ,\rho _{1} ( \mu ) \} \), and the proof is complete. □
Theorem 3.2
Let \(0\leq \mu <1\), \(\alpha \geq 1\), \(\beta \geq 1\). If \(\frac{\varGamma ( \alpha +\beta ) }{\varGamma ( \beta ) } \geq 2 ( 2-\mu ) \), then \(E_{\alpha ,\beta } ( z ) \) is pre-starlike of order μ in \(\mathcal{U}\).
Proof
Consider \(T_{i} ( \rho ,\mu ) ,i=1,2,3\), as in Theorem 3.1. Replacing ρ by μ, we get
It is noticed that, for \(0\leq \mu <1\),
Similarly,
Therefore, \(T_{1} ( \mu ) \) is maximum. Hence
This is equivalent to
□
Corollary 3.3
Let \(\alpha \geq 1\), \(\beta \geq 1\). Then \(E_{\alpha ,\beta } ( z ) \in \mathcal{C}\) if \(\frac{\varGamma ( \alpha +\beta ) }{\varGamma ( \beta ) }\geq 4\).
Proof
It is noticed that, for \(\mu =0\), we have \(\frac{z}{ ( 1-z ) ^{2}}\ast f ( z ) \in \mathcal{S}^{\ast }\). By using the definition of convolution, it is easy to see that \(zf^{\prime } ( z ) \in \mathcal{S}^{\ast }\). Therefore, by Alexander relation it follows that \(f\in \mathcal{C}\). We also see from Theorem 3.2 that
Now
similarly,
Therefore, \(T_{1} ( 0 ) \) is maximum. Hence
This shows that \(E_{\alpha ,\beta } ( z ) \in \mathcal{C}\). □
Example 3.4
For \(\alpha =3\), \(\beta =1\), we have \(\frac{\varGamma ( \alpha + \beta ) }{\varGamma ( \beta ) }\geq 4\), therefore the function
is in \(\mathcal{C}\).
Example 3.5
For \(\alpha =1\), \(\beta =4\), we have \(\frac{\varGamma ( \alpha + \beta ) }{\varGamma ( \beta ) }=4\), therefore the function
is in \(\mathcal{C}\).
The mappings of these functions are given in Fig. 1.
Corollary 3.6
Let \(\alpha \geq 1\), \(\beta \geq 1\). Then \(E_{\alpha ,\beta } ( z ) \in \mathcal{S}^{\ast } ( \frac{1}{2} ) \) if \(\frac{\varGamma ( \alpha +\beta ) }{\varGamma ( \beta ) }\geq 3\).
Proof
Consider \(T_{i} ( \rho ,\mu ) \), \(i=1,2,3\), as in Theorem 3.1. Replacing ρ by μ, we get
For \(\mu =\frac{1}{2}\),
Now
Similarly,
Therefore, \(T_{1} ( \frac{1}{2} ) \) is maximum. Hence
Therefore, for \(\alpha \geq 1\), \(\beta \geq 1\), and \(\frac{\varGamma ( \alpha +\beta ) }{\varGamma ( \beta ) } \geq 3\), \(E_{\alpha ,\beta } ( z ) \in \mathcal{S}^{\ast } ( \frac{1}{2} ) \). □
Example 3.7
For \(\alpha =1\), \(\beta =3\), we have \(\frac{\varGamma ( \alpha + \beta ) }{\varGamma ( \beta ) }=3\), therefore the function
is in \(\mathcal{S}^{\ast } ( \frac{1}{2} ) \).
Example 3.8
For \(\alpha =2\), \(\beta =2\), we have \(\frac{\varGamma ( \alpha + \beta ) }{\varGamma ( \beta ) }=6\), therefore the function
is in \(\mathcal{S}^{\ast } ( \frac{1}{2} ) \).
The mappings of these functions are given in Fig. 2.
Theorem 3.9
Let \(\mu \geq 1\), \(\eta \geq \frac{-\mu +\sqrt{\mu ^{2}+4\mu -4}}{2}\) with \(\alpha \geq 1\), \(\beta \geq 2\) If \(M_{1}=(1+\eta )(1-\mu - \eta )>0\) and \(M_{2}=+2\mu +\eta -1>0\), then \(E_{\alpha ,\beta }(z)\) is starlike of order \(2\mu +\eta -1\).
Proof
It is observed that \(( E_{\alpha ,\beta } ) _{n}(z)= \sum_{k=1}^{\infty }a_{k}z^{k}\) gives \(a_{1}=1\) and \(a_{k}=\frac{ \varGamma ( \beta ) }{\varGamma ( \alpha ( k-1 ) +\beta ) }\) for \(k\geq 2\). The relation between \(a_{k}\) and \(a_{k+1}\) is
To prove this theorem, it is enough to show that \(\{ a_{k} \} \) satisfies conditions (2.1) and (2.3) of Lemma 2.4. Using the above relation and simple computations yields
where \(h(k)\) is defined as follows:
It is observed that under the conditions \(\mu \geq 1\), \(\eta \geq \frac{- \mu +\sqrt{\mu ^{2}+4\mu -4}}{2}\), \(\alpha \geq 1\), and \(\beta \geq 3\), expression (3.2) is positive for \(k\geq 1\). It remains to verify (2.3). That is,
Clearly,
where \(g(k)\) is defined as follows:
It is observed that under the conditions \(\mu \geq 1\), \(\eta \geq \frac{- \mu +\sqrt{\mu ^{2}+4\mu -4}}{2}\), \(2\mu +\eta >1\), \(\alpha \geq 1\), and \(\beta \geq 2\), expression (3.3) is positive for \(k\geq 1\), which completes the proof. □
Theorem 3.10
Let \(\mu \geq 1\), \(\eta \geq \frac{-\mu +\sqrt{\mu ^{2}+4\mu -4}}{2}\), \(2\mu +\eta >1\), \(\alpha \geq 1\), \(\beta \geq 2\), \(a_{1}=1\), \(a_{k}\geq 0\) satisfy
for \(k\geq 4\). Then \(( E_{\alpha ,\beta } ) _{n}(z)= \sum_{k=4}^{n}a_{k}z^{k}\) is convex in the direction of imaginary axis.
Proof
To show that Mittag-Leffler function is convex in the direction of imaginary axis, we will prove that \(z ( E_{\alpha ,\beta } ) _{n}^{\prime }(z) \) is a typically real function. Also \(( E_{ \alpha ,\beta } ) _{n}(z)\) has real coefficients. Set
where \(b_{k}=\frac{\varGamma ( \beta ) }{\varGamma ( \alpha ( k-1 ) +\beta ) }\). To get the result, it is required that \(\{ b_{k} \} \) must satisfy the conditions mentioned in Lemma 2.3. Consider
for \(k=1,2,3,\ldots,n-1\). Take
here \(q(k)\) is defined as
Since Γ is an increasing function in \([ \frac{3}{2}, \infty )\), therefore (3.5) becomes positive when \(\mu \geq 1\), \(\eta \geq \frac{-\mu +\sqrt{\mu ^{2}+4 \mu -4}}{2}\), \(2\mu +\eta >1 \), \(\alpha \geq 1\), and \(\beta \geq 2\). Thus \(\{ b_{k} \} \) satisfies the conditions of Lemma 2.3. Therefore, by using the minimum principle for harmonic functions under the conditions \(\mu +\eta \in ( 0,\frac{1+\eta }{2} ] \),
and
The Schwarz reflection principle yields that \(\operatorname{Im} ( zf_{n}^{ \prime } ( z ) ) <0\) for \(\theta \in ( \pi ,2 \pi ) \). So \(zf_{n}^{\prime } ( z ) \) is a typically real function, which is equivalent to saying that \(f_{n}(z)\) is convex in the direction of imaginary axis. □
Theorem 3.11
Let \(\mu \in \mathbb{R} \) and \(\eta \geq 0\) such that \(2\mu +\eta >1\), and let \(a_{1}=1\) and \(a_{k}\geq 0\) satisfy
and
If \(\alpha \geq 1\) and \(\beta \geq 2\), then \(( E_{\alpha , \beta } ) _{n}(z)=z+\sum_{k=2}^{n}a_{k}z^{k}\) satisfies \(\operatorname{Re}(f_{n}^{\prime }(z))>1-\frac{\mu +\eta }{\mu _{0}^{\ast }}\).
Proof
Let \(\sigma -1=-\frac{\mu +\eta }{\mu _{0}^{\ast }}\) and \(( E _{\alpha ,\beta } ) _{n}(z)=z+\sum_{k=2}^{n}\frac{\varGamma ( \beta ) }{\varGamma ( \alpha ( k-1 ) + \beta ) }z^{k}\), where \(a_{k}=\frac{\varGamma ( \beta ) }{\varGamma ( \alpha ( k-1 ) +\beta ) }\). Then
where \(c_{k}=\frac{(k+1)b_{k+1}}{1-\sigma } \) and \(c_{0}=1\) for \(1 \leq k\leq n-1\). It is observed that under the conditions \(\alpha \geq 1\) and \(\beta \geq 2\) the coefficients \(a_{k}\) are positive. Therefore, \(c_{k}>0\) for \(k\geq 1\). To prove this theorem, we will show that the coefficients \(\{ c_{k} \} \) are decreasing and satisfy (2.5). Now, for this, consider
for \(k=1,2,3,\ldots,n-2\). This shows that the coefficients of Mittag-Leffler function are decreasing and \(c_{1}< c_{0}\Rightarrow 2b _{2}<1-\sigma \). Now to have (2.5), consider
It is clear that the above relation is positive for \(n\geq k\), \(\alpha \geq 1 \), and \(\beta \geq 2\). Also, Γ is an increasing function in \([ \frac{3}{2},\infty ) \). This yields (3.7). Using similar arguments and the minimum principle for harmonic function gives the required result. □
4 Conclusion
In this paper, we have studied the normalized Mittag-Leffler function of two parameters. We have investigated new properties including pre-starlikeness, convexity, and starlikeness of order \(1/2\). Sufficient conditions for the normalized Mittag-Leffler function to be starlike of order \(2\mu +\eta -1\) have also been studied. Moreover, we have found the convexity of Mittag-Leffler functions in the direction of imaginary axis.
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The work here is partially supported by HEC grant:5689/Punjab/NRPU/R&D/HEC/2016.
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MR and SN came up with the main thoughts, SN proved the main theorems, SNM reviewed them. All authors read and approved the final manuscript.
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Noreen, S., Raza, M. & Malik, S.N. Certain geometric properties of Mittag-Leffler functions. J Inequal Appl 2019, 94 (2019). https://doi.org/10.1186/s13660-019-2044-4
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DOI: https://doi.org/10.1186/s13660-019-2044-4
MSC
- 33E12
- 30C45
Keywords
- Mittag-Leffler functions
- Starlike functions
- Close-to-convex functions
- Typically real function
- Convex functions in the direction of imaginary axis
- Pre-starlike functions