Abstract
Cotîrlă and Szász (Comput Methods Funct Theory: 2023) solved a conjecture related with inclusion relation for Bessel functions, proposed by Baricz and András (Complex Var Elliptic Equ 54(7): 689–696, 2009). In this paper, we prove this conjecture for normalized Struve functions by using subordination factor sequences.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Let \(\mathbb {D}=\{z\in \mathbb {C}:|z|<1\}\) be the open unit disk. An analytic function f in \(\mathbb {D}\) with \(0 \ne f^\prime \left( 0\right) \) is convex, i.e. \({\text {Re}}\left[ 1+zf^{\prime \prime }\left( z\right) /f^\prime \left( z\right) \right] >0\) for all \(z\in \mathbb {D}, \) if and only if f is univalent in \(\mathbb {D}\) with \(f\left( \mathbb {D}\right) \) being convex.
The Struve functions, denoted by \(H_\nu \), are solutions of the inhomogeneous Bessel differential equation
and they have the power series form
We define the normalized form of Struve functions by the equality
An other normalized form of \(H_\nu \), which we are going to study is
After Louis de Branges proved the Bieberbach Conjecture by using the generalized hypergeometric function in 1984, special functions became popular in studies of geometric function theory. Recently, there has been great interest dealing with various properties of special functions such as Bessel, Struve and Lommel functions of the first kind [1,2,3,4,5,6,7,8,9,10, 12, 14, 15].
The aim of the paper is to prove the implications
regarding normalized Struve functions. In order to prove our result we need the definitions and lemmas presented in the next section.
2 Preliminaries
In this section, we are devoted to giving and proving our main results.
Definition 2.1
Let f, g and h be analytic functions in \(\mathbb {D}\). If the function h satisfies the conditions \(h(0)=0\), \(|h(z)|<1, \ z\in \mathbb {D}\) and \(f(z)=g(h(z)), \ z\in \mathbb {D}\), then we say that the function f is subordinate to the function g.
This subordination is denoted by \(f\prec {g}\).
If g is univalent and \(f(0)=g(0)\), then the following equivalence holds:
Definition 2.2
An infinite sequence \((b_n)_{n\ge 1}\) of complex numbers will be called a subordination factor sequence if for every convex function f defined by \(f(z)=\sum _{n=1}^\infty {a_n}z^n\) we have \(g\prec {f}\), where g is defined by \(g(z)=\sum _{n=1}^\infty {a_nb_n}z^n, z\in \mathbb {D}\).
Definition 2.3
Let f and g be two analytic functions on \(\mathbb {D}\) defined by the power series \(f(z)=\sum ^\infty _{n=1}a_nz^n\) and \(g(z)=\sum ^\infty _{n=1}b_nz^n\). The convolution of the functions f and g denoted by \(f*g\) is defined by the equality
Remark 2.1
Definition 2.1 and 2.2 can be reformulated using convolution as follows: Let \((b_n)_{n\ge 1}\) be a sequence of complex numbers and the function \(\phi \) be defined by the equality \(\phi (z)=\sum ^\infty _{n=1}b_nz^n\). An infinite sequence \((b_n)_{n\ge 1}\) of complex numbers will be called a subordination factor sequence if for every convex function f we have
Lemma 2.1
[16] The following two properties of sequence of complex numbers are equivalent:
-
(I)
The infinite sequence \((b_n)_{n\ge 1}\) of complex numbers is a subordination factor sequence.
-
(II)
The inequality \(\frac{1}{2}+{\text {Re}}\sum _{n=1}^\infty {b_n}z^n>0\), holds for every \(z\in \mathbb {D}\).
Lemma 2.2
[11] Let \((f_n)_{n\ge 0}\) be a sequence of real numbers such that \(f_0=1\), \(f_n-2f_{n+1}+f_{n+2}\ge 0\) and \(f_{n}-f_{n+1} \ge 0\) for all \(n\in \{0,1,2,3,\ldots \}\), then the inequality holds
In Theorem 2 of the study [13], the authors show that for \( n\ge 1\) and \(z\in \mathbb {D}\), if \(f(z)=z+a_{n+1}z^{n+1}+\dots \) and \(|f^{\prime \prime }(z)|\le \frac{n}{n+1}\), then f is convex. Then, the following lemma is a consequence of this result.
Lemma 2.3
[13] If f is of the form \(f(z)=\sum ^\infty _{n=1}a_nz^n\) with \(a_1\ne 0\) and
then f is univalent with \(f(\mathbb {D})\) convex set in \(\mathbb {C}\).
The following lemmas are the key tool in the proof of our main results.
Lemma 2.4
If \(y>x>\frac{5}{2}\) then the inequality holds
Proof
The inequality we have to prove is equivalent to
In order to prove this inequality we will prove that the function \(u:(\frac{3}{2},\infty )\rightarrow \mathbb {R}, \ u(t)=\frac{1+t}{\Gamma (1+t)}\) is decreasing.
We have
The sum \(\sum _{n=1}^\infty \frac{t}{n(n+t)}\) is increasing with respect to t and \(t>1\). Thus it follows
Consequently the mapping \(u:(\frac{3}{2},\infty )\rightarrow \mathbb {R}, \ u(t)=\frac{1+t}{\Gamma (1+t)}\) is decreasing. On the other hand the condition \(y>x>\frac{5}{2}\) is equivalent to \(y-1>x-1>\frac{3}{2}\) and the monotony of u implies \(u(y-1)\le {u}(x-1)\) and this inequality is equivalent to \(\frac{x}{\Gamma (x)}\ge \frac{y}{\Gamma (y)}\). The proof is completed \(\square \)
Lemma 2.5
Provided that \(\mu>\nu >0\) the inequality
holds for every \(z\in \mathbb {D}\).
Proof
We use Lemma 2.2 and Lemma 2.4 in this proof. We have \(f_0=1\) and \(f_n=\frac{\Gamma (n+\nu +\frac{3}{2})}{\Gamma (n+\mu +\frac{3}{2})}\) in case \(n\ge 1\). Thus \(f_0-f_1=1-\frac{\Gamma (\nu +\frac{5}{2})}{\Gamma (\mu +\frac{5}{2})}\ge 0\) because \(\mu>\nu >0\) and if \(n\ge 1\), then
In order to prove the inequality \(f_n-2f_{n+1}+f_{n+2}\ge 0\), we distinguish two cases exactly as before. In case \(n=0\), we have to show \(f_0-2f_{1}+f_{2}\ge 0\).
We have
On the other hand Lemma 2.4 implies \(\sqrt{\frac{\nu +\frac{5}{2}}{\mu +\frac{5}{2}}}-\sqrt{\frac{\Gamma (\nu +\frac{5}{2})}{\Gamma (\mu +\frac{5}{2})}}\ge 0\). Thus we get \(f_0-2f_{1}+f_{2}\ge 0\).
The proof of the inequality \(f_n-2f_{n+1}+f_{n+2}\ge 0\) in case \(n\ge 1\) is much easier than the particular case for \(n=0\).
We have
Thus, the proof is completed. \(\square \)
Lemma 2.6
Provided that \(\mu>\nu >0\) the inequality holds
Proof
We use for the second time 2.2 in order to prove this lemma. This time \(f_0=1\) and \(f_n=\frac{\Gamma (\nu +1+\frac{n}{2})}{\Gamma (\mu +1+\frac{n}{2})}\).
Since \(\mu>\nu >0\) it follows that \(f_0=1>\frac{\Gamma (\nu +1+\frac{1}{2})}{\Gamma (\mu +1+\frac{1}{2})}=f_1\). We have to show \(f_n=\frac{\Gamma (\nu +1+\frac{n}{2})}{\Gamma (\mu +1+\frac{n}{2})}>\frac{\Gamma (\nu +1+\frac{n+1}{2})}{\Gamma (\mu +1+\frac{n+1}{2})}=f_{n+1}\) in case \(n\ge 1\). This inequality can be rewritten as follows
We define the function \(u:(1,\infty )\rightarrow (0,\infty )\) by \(u(x)=\frac{\Gamma (x)}{\Gamma (x+\frac{1}{2})}\). We have
B(x, y) denotes the Euler’s Beta function defined by \(B(x,y)=\int _0^1t^{x-1}(1-t)^{y-1}dt\). Thus equality (2.5) implies that the mapping u is strictly decreasing and consequently \(u(\nu +1+\frac{n}{2})>u(\mu +1+\frac{n}{2})\), which is equivalent to (2.4). In order to finish the proof we have to show that
We define the mapping \(v:(1,\infty )\rightarrow (0,\infty )\) by \(v(x)=\frac{\Gamma (x)}{\Gamma (x+\mu -\nu )}=\frac{B(x,\mu -\nu )}{\Gamma (\mu -\nu )}\). The equivalence holds
if and only if
Since \(f_0=1\ge {v}(\nu +1)\), it follows that in order to prove (2.6) it is enough to prove
On the other hand we have \(v''(x)=\frac{1}{\Gamma (\mu -\nu )}\int _0^1t^{x-1}\ln ^2(t)(1-t)^{\mu -\nu -1}dt\ge 0\). Thus the function v is convex and consequently inequality (2.7) hods. \(\square \)
3 The Main Result
Our first main result is the following theorem, which gives the monotonicity result for the function \(h_\nu \).
Theorem 3.1
If \(\mu>\nu >\nu _0=\frac{\sqrt{8585}}{90}-\frac{5}{3}\approx -0.63716\), then the following inclusion holds
where \(\nu _0\) is the biggest real root of the equation \(1620\nu ^2+5400\nu +2783=0\) and \(h_\nu \) is defined by (1.1).
Proof
We have
Since the following inequalities hold
and
we get
Thus Lemma 2.3 implies that \(h_\nu \) is a convex function. According to Lemma 2.5 the sequence \(\left( \frac{\Gamma (n+\nu +\frac{3}{2})}{\Gamma (n+\mu +\frac{3}{2})}\right) _{n\ge 1}\) is a subordination factor sequence and the function \(h_\nu \) is convex, consequently the subordination follows
where \(\chi (z)=\sum ^\infty _{n=1}\frac{\Gamma (n+\nu +\frac{3}{2})}{\Gamma (n+\mu +\frac{3}{2})}z^n\). Since \(h_\mu (0)=h_\nu (0)\), the subordination \(h_\mu \prec {h_\nu }\) is equivalent to \(h_\mu (\mathbb {D})\subset {h_\nu }(\mathbb {D})\). Thus, the proof is completed. \(\square \)
Our second result is stated as follows.
Theorem 3.2
If \(\mu>\nu >\nu _1=\frac{\sqrt{28385}}{90}-1\approx 0.87198\), then the following inclusion holds
where \(\nu _1\) is the biggest real root of the equation \(1620\nu ^2+3200\nu -4057=0\) and \(g_\nu \) is defined by (1.2).
Proof
We have
After simple calculations, the inequalities
and
true for \(\ n\ge 2, \ \nu >\nu _1\). Finally
Thus, we get
Thus Lemma 2.3 implies that \(g_\nu \) is a convex function. According to Lemma 2.6 the sequence \(\left( \frac{\Gamma (\nu +1+\frac{n}{2})}{\Gamma (\mu +1+\frac{n}{2})}\right) _{n\ge 1}\) is a subordination factor sequence and the function \(g_\nu \) is convex, consequently the subordination follows
where \(\psi (z)=\sum ^\infty _{n=1}\frac{\Gamma (\nu +1+\frac{n}{2})}{\Gamma (\mu +1+\frac{n}{2})}z^n\). Since \(g_\mu (0)=g_\nu (0)\), the subordination \(g_\mu \prec {g_\nu }\) is equivalent to \(g_\mu (\mathbb {D})\subset {g_\nu }(\mathbb {D})\). Thus, the proof is completed. \(\square \)
References
Alzahrani, R., Mondal, S.R.: Redheffer-type bounds of special functions. Mathematics 11(2), 379 (2023)
Aktaş, İ, Baricz, Á., Orhan, H.: Bounds for radii of starlikeness and convexity of some special functions. Turkish J. Math. 42(1), 211–226 (2018)
Baricz, Á., Ponnusamy, S., Singh, S.: Turán type inequalities for Struve functions. J. Math. Anal. Appl. 445, 971–984 (2017)
Baricz, Á., Yağmur, N.: Geometric properties of some Lommel and Struve functions. Ramanujan J. 42(2), 325–346 (2017)
Baricz, Á., Dimitrov, D.K., Orhan, H., Yağmur, N.: Radii of starlikeness of some special functions. Proc. Amer. Math. Soc. 144(8), 3355–3367 (2016)
Baricz, Á., András, S.: Monotony property of generalized and normalized Bessel functions of complex order. Complex Var. Elliptic Equ. 54(7), 689–696 (2009)
Bohra, N., Ravichandran, V.: Radii problems for normalized Bessel functions of first kind. Comput. Methods Funct. Theory 18(1), 99–123 (2018)
Cotîrlă, L.I., Szász, R.: On the monotony of Bessel functions of the first kind. Comput. Methods Funct. Theory (2023). https://doi.org/10.1007/s40315-023-00498-0
Çetinkaya, A., Cotîrlă, L.I.: Briot-Bouquet differential subordinations for Analytic functions involving the Struve function. Fractal and Fractional. 6(10), 540 (2022)
Deniz, E., Kazımoğlu, S., Çağlar, M.: Radii of uniform convexity of Lommel and Struve functions. Bull. Iranian Math. Soc. 47(5), 1533–1557 (2021)
Fejér, L.: Trigonometrische Reihen und Potenzreihen mit mehrfach monotoner Koeffizientenfolge. Trans. Amer. Math. Soc. 39(1), 18–59 (1936)
Kazımoğlu, S., Gangania, K.: Radius of \(\gamma -\)spirallikeness of order \(\alpha \) of some special functions. Complex Anal. Its Synerg. 9(4), 14 (2023)
Mocanu, P.T.: Some simple criteria for starlikeness and convexity. Libertas Math. 13(1), 27–40 (1993)
Naz, A., Nagpal, S., Ravichandran, V.: Exponential starlikeness and convexity of confluent hypergeometric. Lommel and Struve functions. Mediterr. J. Math. 17, 1–22 (2020)
Noreen, S., Raza, M., Deniz, E., Kazımoğlu, S.: On the Janowski class of generalized Struve functions. Afr. Mat. 30, 23–35 (2019)
Wilf, H.S.: Subordinating factor sequences for convex maps of the unit circle. Proc. Amer. Math. Soc. 12(5), 689–693 (1961)
Funding
Open access funding provided by the Scientific and Technological Research Council of Türkiye (TÜBİTAK).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Dan Volok.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Deniz, E., Szász, R. On The Monotony of Struve Functions. Complex Anal. Oper. Theory 18, 120 (2024). https://doi.org/10.1007/s11785-024-01563-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11785-024-01563-9