Abstract
We consider the convolution operator
on the class of analytic functions \(f(z)=z+a_2z^2+\cdots \), \(|z|<1\), in the complex plane, where \(\zeta \) is complex, \(|\zeta |\le 1\). For \(\zeta =1\), the operator becomes the derivative \(f'(z)\), while for real \(\zeta =q\), \(0<q<1\) we obtain the Jackson’s q-derivative \(\mathrm{d}_qf(z)\).
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Let \({\mathcal {H}}\) denote the class of analytic functions in the unit disc \({{\mathbb {D}}}=\{z:\ |z|<1\}\) on the complex plane \({\mathbb {C}}\). We will use the following notations:
Let the function \(f\in {\mathcal {H}}\) be univalent in the unit disc \({{\mathbb {D}}}\) with the normalization \(f(0)=0\). Then f maps \({\mathbb {D}}\) onto a starlike domain with respect to \(w_0=0\) if and only if
Such function f is said to be starlike in \({\mathbb {D}}\) with respect to \(w_0=0\) (or briefly starlike). Recall that a set \(E\subset {\mathbb {C}}\) is said to be starlike with respect to a point \(w_0\in E\) if and only if the linear segment joining \(w_0\) to every other point \(w\in E\) lies entirely in E, while a set E is said to be convex if and only if it is starlike with respect to each of its points, that is, if and only if the linear segment joining any two points of E lies entirely in E. A function f maps \({\mathbb {D}}\) onto a convex domain E if and only if
and then f is said to be convex in \({\mathbb {D}}\) (or briefly convex). It is well known that if an analytic function f satisfies (1.2) and \(f(0)=0,\)\(f'(0)\ne 0,\) then f is univalent and starlike in \({\mathbb {D}}\). Let \({\mathcal {A}}\) denote the subclass of \({\mathcal {H}}\) consisting of functions normalized by \(f(0)=0\), \(f'(0)=1\). The set of all functions \(f\in {\mathcal {A}}\) that are starlike univalent in \({{\mathbb {D}}}\) will be denoted by \({\mathcal {S}}^*\). The set of all functions \(f\in {\mathcal {A}}\) that are convex univalent in \({\mathbb {D}}\) by \({\mathcal {K}}\). It is known that for \(f\in \mathcal {A}\), condition (1.3) is sufficient for starlikeness of f. Also the condition
is sufficient for starlikeness of f. In this paper we shall consider certain sufficient conditions for starlikeness of order 1 / 2. The class \(\mathcal {S}^*(\alpha )\) of starlike functions of order \(\alpha <1\) may be defined as
The class \({\mathcal {S}}^*(\alpha )\) and the class \({\mathcal {K}}(\alpha )\) of convex functions of order \(\alpha < 1\)
were introduced by Robertson in [10]. It is known the old Strohhäcker result [16] that \( {\mathcal {K}}(0)\subset {\mathcal {S}}^*(\alpha )\subset {\mathcal {S}}^*(0)\). Furthermore, note that if \(f\in \mathcal {K}(\alpha )\) then \(f\in \mathcal {S}^*(\delta (\alpha ))\), see [17], where
Robertson [11] proved that if \(f\in \mathcal {A}\) with \(f(z)/z\ne 0\) and if there exists a k, \(0<k\le 2\), such that
then \(f(z)\in {\mathcal {S}}^*(2/(2+k))\). In [8], it was proved that for \(f\in \mathcal {A}\) with \(f(z)f'(z)/z\ne 0,\) if
then \(f(z)\in {\mathcal {S}}^*\). Several more complicated sufficient conditions for starlikeness and for convexity are collected in the book [7], Chap. 5. Recall also, that if \(f\in \mathcal {A}\) satisfies
for some \(g\in \mathcal {S}^*\) and some \(\alpha \in (-\pi /2,\pi /2)\), then f is said to be close-to-convex in \(\mathbb {D}\) and denoted by \(f\in \mathcal {C}\). An univalent function \(f\in \mathcal {A}\) belongs to \(\mathcal {C}\) if and only if the complement E of the image-region \(F=\left\{ f(z): |z|<1\right\} \) is the union of rays that are disjoint (except that the origin of one ray may lie on another one of the rays).
On the other hand, if \(f\in \mathcal {A}\) satisfies
for some \(g\in \mathcal {S}^*\) and some \(\beta \in [0,1]\), then f is said to be a Bazilevic̆ function of type \(\beta \) and denoted by \(f\in \mathcal {B}(\beta )\).
Jackson in [5, 6] introduced and studied the q-derivative, \(0<q<1\), as
and \(d_qf(0)=f'(0)\). Then
where
Making use of q-derivative, Argawal and Sahoo in [1] introduced the class \({\mathcal {S}}^*_q(\alpha )\). A function \(f\in \mathcal {A}\) is said to belong to the class \({\mathcal {S}}^*_q(\alpha )\), \(0\le \alpha <1\), if
If \(q\rightarrow 1^-\), the class \({\mathcal {S}}^*_q(\alpha )\) reduces to the class \({\mathcal {S}}^*(\alpha )\). If \(\alpha =0\), the class \({\mathcal {S}}^*_q(\alpha )\) coincides with the class \({\mathcal {S}}^*_q(0)={\mathcal {S}}^*_q\), which was first introduced by Ismail et al. in [3] and was considered in [2, 9, 12, 14].
Let \(\mathcal {H}\) denote the class of analytic functions in the open unit disc \({\mathbb {D}}=\{z:\ |z|<1\}\) on the complex plane \(\mathbb {C}\). Also, let \( {\mathcal {A}}\) denote the subclass of \(\mathcal {H}\) comprising of functions f normalized by \(f(0)=0\), \(f'(0)=1,\) and let \(\mathcal {S}\subset \mathcal {A}\) denote the class of functions which are univalent in \({\mathbb {D}}\).
2 q-derivative operator on convex functions
Theorem 2.1
If f is in the class \(\mathcal {K}\) of convex univalent functions, then for all q, \(0<q<1\), we have
is in the class \(\mathcal {S}^*\left( \frac{1-q}{2(1+q)}\right) \) of starlike univalent functions of order \(\frac{1-q}{2(1+q)}\).
Proof
It is easy to check that for each \(q\in \mathbb {C}\), \(|q|\le 1\), \(q\ne 1\), the function
is convex univalent in \(\mathbb {D}\), see also [4, Th.17,p.170]. Hence,
is starlike univalent in \(\mathbb {D}\). With a little more effort, we can find that \(zh'_q(z)\) is starlike of order \(\frac{1-q}{2(1+q)}\). On the other hand, we have that
where \(*\) denotes the Hadamard product, or convolution, of power series. Hence, \(z\mathrm{d}_qf(z)\) is a convolution of f(z) with a starlike function of order \(\frac{1-q}{2(1+q)}\). Because of the famous result [13] that \(\mathcal {K} *\mathcal {S}^*(\alpha )=\mathcal {S}^*(\alpha )\), we finally obtain that the function in (2.1) is in the class \(\mathcal {S}^*\left( \frac{1-q}{2(1+q)}\right) \). \(\square \)
The known Alexander theorem says that
Therefore, Theorem 2.1 is an equivalent of Alexander theorem
It is known that
A question is: Is it true that
In terms of the convolution, this problem we may write as: exists there a q, \(0<q<1\), that for given \(f(z)=z+a_2z^2+\cdots \in \mathcal {S}^*\), we have
The answer on the question (2.5) is: no. Namely, for the starlike function
\(a_n=n\) and the function in (2.5) becomes
which is not in the class \(\mathcal {K}\) because it has the coefficients \(n/[n]_{q}\) greater than 1, otherwise than in \(\mathcal {K}\).
Because the Koebe function \(f(z)=z/(1-z)^2\in \mathcal {C}\), we have for the class of close-to-convex functions \(\mathcal {C}\), (1.5), that
is also false. The above facts we can write \(\mathcal {S}^*\not \subset zd_q(\mathcal {K})\), \(\mathcal {C}\not \subset zd_q(\mathcal {K})\), or equivalently \(\mathcal {B}(0)\not \subset zd_q(\mathcal {K})\), \(\mathcal {B}(1)\not \subset zd_q(\mathcal {K})\). A question for \(\mathcal {B}(\beta )\) is: What about \(\beta \in (0,1)\)? So we have the following problem.
Open Problem. There exists a \(\beta \), \(\beta \in (0,1)\), such that
where \(\mathcal {B}(\beta )\) is the class of Bazilevic̆ functions of type \(\beta \).
3 \(\zeta \)-derivative operator
It is easy to check that if \(q\rightarrow 1^-\) the function \(zh'_q(z)\) (2.3) becomes the well-known Koebe function
For each \(f\in \mathcal {A}\), we can express its derivative in terms of the Koebe function as
It is a natural to consider a generalization of (3.2) for \(\zeta \in \mathbb {C}\), \(|\zeta |\le 1\):
For \(\zeta =1\), we have the derivative \(f'\), while for \(\zeta =q\), \(0<q<1\) we obtain the Jackson’s q-derivative of f, namely \(\mathrm{d}_qf\), which is defined in (1.7). Therefore, for
we have
where
For these reasons, we can look on q-derivative \(\mathrm{d}_qf\) as a special case of the convolution operator (3.4).
Corollary 3.1
If f is in the class \(\mathcal {K}\) of convex univalent functions, then for all \(\zeta \), \(|\zeta |\le 1\), we have
is in the class \(\mathcal {S}^*\) of starlike univalent functions.
Proof
The proof runs in the same way as the proof of Theorem 2.1, because the function
is starlike for all complex \(\zeta \), \(|\zeta |<1\). \(\square \)
Theorem 3.2
If f is in the class \(\mathcal {K}\) of convex univalent functions, then for all \(\zeta \), \(|\zeta |\le 1\), we have
or
Proof
It is known [13, 15, p.10], that if \(f\in \mathcal {K}\), then for all z, v, and \(w\in \mathbb {D}\), we have
If we put \(w=\zeta z\) and \(v=\zeta ^2z\) in (3.8), then we obtain
Trivial calculations give
or
This establishes (3.6) and (3.7). \(\square \)
Corollary 3.3
If f is in the class \(\mathcal {K}\) of convex univalent functions, then for all q, \(0<q< 1\), we have
or
Theorem 3.4
If f is in the class \(\mathcal {K}\) of convex univalent functions, then for all \(\zeta \), \(|\zeta |\le 1\), we have
or
Proof
From (3.8), we have that for \(f\in \mathcal {K}\), for all t, v, and \(w\in \mathbb {D}\), we have
If we put \(v=z\), \(w=\zeta z\) and \(t=\zeta ^2z\) in (3.11), then we obtain
After some calculations, we obtain
or
This proves (3.9) and (3.10). \(\square \)
Corollary 3.5
If f is in the class \(\mathcal {K}\) of convex univalent functions, then for all q, \(0<q< 1\), we have
or
Corollary 3.3 and Corollary 3.5 may be applied to obtain a bound for the modulus.
Corollary 3.6
If f(z) is in the class \(\mathcal {K}\) of convex univalent functions, then for all q, \(0<q< 1\), we have
4 \(\zeta \)-starlike functions of order \(\alpha \)
Definition
Let f be in \(\mathcal {A}\). For given \(\zeta \), \(|\zeta |\le 1\), we say that f is in the class \(\mathcal {S}^*(\zeta ,\alpha )\) of \(\zeta \)-starlike functions of order \(\alpha \), \(\alpha \in [0,1)\) if
where the operator \(\mathrm{d}_{\zeta }f\) is defined in (3.3).
Remark
For \(\zeta =1\), condition (4.1) becomes
and the class \(\mathcal {S}^*(1,\alpha )\) becomes the well-known class of starlike functions of order \(\alpha \). For \(\zeta \ne 1\), condition (4.1) becomes
Theorem 4.1
If f is in the class \(\mathcal {K}\) of convex univalent functions, then \(f(z)\in \mathcal {S}^*(\zeta ,1/2)\) for each \(|\zeta |\le 1\).
Proof
For \(\zeta =1\), Theorem 4.1 becomes the known fact that each convex function is starlike of order 1 / 2. Now, let \(\zeta \ne 1\). If we put in (3.8) \(v=0\), next write \(z=\zeta z\), and then, \(w=z\); then, we obtain
From (4.3), we obtain that \(f\in \mathcal {S}^*(\zeta ,1/2)\) for each \(|\zeta |\le 1\). \(\square \)
Theorem 4.2
If f is in the class \(\mathcal {K}\) of convex univalent functions, then
for each \(|\zeta |<1\).
Proof
For \(v=\zeta z\), the condition (3.8) becomes
For \(w\rightarrow z\), this gives
and finally we get (4.4). \(\square \)
Corollary 4.3
If f is in the class \(\mathcal {K}\) of convex univalent functions, then
for each \(0<q<1\).
It is a natural question whether
is in the class of \(\zeta \)-starlike functions \(\mathcal {S}^*(\zeta ,\alpha )\) defined in (4.1), for some \(\alpha \).
We answer this question in the following theorem.
Theorem 4.4
If \(|\zeta |\le 1\), then the function (4.6) is in the class \(\zeta \)-starlike functions \(\mathcal {S}^*(\zeta ,\alpha )\), where
Proof
From (3.3), the function (4.6) is in the class \(\zeta \)-starlike functions \(\mathcal {S}^*(\zeta ,\alpha )\) if and only if
where
We have
An elementary calculation shows that the function \(F(\zeta ,z)=\frac{1+\zeta z}{1-\zeta ^2z}\) maps the unit disc \(\mathbb {D}\) on a circle with a centre \(s=\frac{1+\zeta \overline{\zeta ^2}}{1-|\zeta |^4}\) and a radius \(r=\frac{|\zeta +\zeta ^2|}{1-|\zeta |^4}. \) So
for all \(|z|<1.\) Thus for all \(\zeta \), \(|\zeta |\le 1\), we have
\(\square \)
Corollary 4.5
If \(|\zeta |\le 1\), then the function (4.6) is in the class \(\zeta \)-starlike functions \(\mathcal {S}^*(\zeta ,0)\).
Corollary 4.6
For real \(\zeta \), \(0<\zeta <1,\) the function (4.6) is in the class \(\zeta \)-starlike functions \(\mathcal {S}^* \left( \zeta , \frac{1-\zeta }{1+\zeta ^2}\right) \).
References
Agrawal, S., Sahoo, S.K.: A generalization of starlike functions of order alpha. Hokkaido Math. J. 46, 15–27 (2017)
Agrawal, S., Sahoo, S.K.: Geometric properties of basic hypergeometric functions. J. Diff. Equ. Appl. 20, 1502–1522 (2014)
Ismail, M.E.H., Merkes, E., Styer, D.: A generalization of starlike functions. Complex Var. 14, 77–84 (1990)
Goodman, A.W.: Univalent Functions, Vols. I and II. Mariner Publishing Co., Tampa (1983)
Jackson, F.H.: On \(q\)-functions and certain difference operator. Trans. R. Soc. Edinb. 46, 253–281 (1908)
Jackson, F.H.: On \(q\)-definite integrals. Q. J. Pure Appl. Math. 41, 193–203 (1910)
Miller, S.S., Mocanu, P.T.: Differential Subordinations, Theory and Applications. Series of Monographs and Textbooks in Pure and Applied Mathematics, vol. 225. Marcel Dekker Inc., New York (2000)
Mocanu, P.T.: On a theorem of Robertson. Babeş-Bolyai Univ., Fac. of Math. Res. Sem., Seminar of Geometric Function Theory 5, 77–82 (1986)
Raghavendar, K., Swaminathan, A.: Close-to-convexity of basic hypergeometric functions using their Taylor coefficients. J. Math. Appl. 35, 111–125 (2012)
Robertson, M.S.: On the theory of univalent functions. Ann. Math. 37, 374–408 (1936)
Robertson, M.S.: Certain classes of starlike functions. Mich. Math. J. 32, 135–140 (1985)
Rønning, F.: A Szegö quadrature Formula Arising from \(q\)-starlike Functions: Continued Fractions and Orthogonal Functions, Theory and Applications, pp. 345–352. Marcel Dekker Inc., New York (1994)
Ruscheweyh, St, Sheil–Small, T.: Hadamard product of schlicht functions and the Poyla-Schoenberg conjecture. Commun. Math. Helv. 48, 119–135 (1973)
Sahoo, S.K., Sharma, N.L.: On a generalization of close-to-convex functions. Ann. Polon. Math. 113, 93–108 (2015)
Schober, G.: Univalent Functions-Selected Topics. Lecture Notes in Math, vol. 478. Springer, Berlin (1975)
Strrohhäcker, E.: Beitrage zür Theorie der Schlichter Functionen. Math. Z. 37, 356–380 (1933)
Wilken, D.R., Feng, J.: A remark on convex and starlike functions. J. Lond. Math. Soc. 21(2), 287–290 (1980)
Author information
Authors and Affiliations
Corresponding author
Additional information
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 distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Piejko, K., Sokół, J. On convolution and q-calculus. Bol. Soc. Mat. Mex. 26, 349–359 (2020). https://doi.org/10.1007/s40590-019-00258-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40590-019-00258-y