Abstract
In this paper, we introduce a new class k-\(\mathcal{US}(q,\gamma ,m,p)\), \(\gamma \in\mathbb{C}\backslash \{0\}\), of multivalent functions using a newly defined q-analogue of a Salagean type differential operator. We investigate the coefficient problem, Fekete–Szego inequality, and some other properties related to subordination. Relevant connections of the results presented here with those obtained in the earlier work are also pointed out.
Similar content being viewed by others
1 Introduction
For a positive integer p, let \(\mathcal{A}_{p}\) denote the set of all functions \(f(z)\) which are analytic and p-valent in the open unit disk \(E=\{z\in\mathbb{C}: \vert z \vert <1\}\) and have series expansion of the form
Also, let \(f\ast g\) denote the convolution (or Hadamard product) of \(f,g\in \mathcal{A}_{p}\) defined as follows:
where \(f(z)\) is given by (1.1) and \(g(z)=z^{p}+\sum_{n=p+1}^{\infty }b_{n}z^{n}\).
Quite recently, q-analysis has influenced the researchers a lot due to rapid applications in mathematics and related fields. In the last century many well-known researchers (for details, see [1, 4, 6–10, 13, 14, 21, 22, 32]) did great work on q-calculus and found numerous applications. It is worth mentioning that convolution theory helps many researchers to investigate a number of properties of analytic univalent and multivalent functions. Several differential and integral operators were defined using ordinary derivative; for details, see [29].
Due to growing applications of q-calculus, investigators are interested in studying properties of functions using q-operators instead of ordinary differential operators; for comprehensive study, we refer to Kanas and Reducanu [15], Mahmood and Darus [19], and Mahmood and Sokol [20]. In this paper we define a q-analogue of a Salagean type operator and study its effect on multivalent functions in conic domains.
For any non-negative integer n, the q-integer number n denoted by \([n]_{q}\) is defined by
For a non-negative integer n, the q-number shift factorial is defined as
We note that when \(q\rightarrow 1\), \([n]_{q}!\) reduces to the classical definition of factorial. In general, \([t]_{q}\) is defined as follows:
For \(f\in A\), in [5], the q-derivative operator or q-difference operator is defined as follows:
It can easily be seen that
Taking motivation from the above mentioned work, we define new convolution operators as follows.
Let
Using the functions \(\varPhi ( p,q,m,z ) \) and the definition of q-derivative along with the idea of convolution, we now define the following differential operator \(\mathcal{S}_{q,p}^{m}f(z):\mathcal{A}_{p}\rightarrow \mathcal{A}_{p}\) for multivalent functions
where
For \(p=1\), the operator \(\mathcal{S}_{q,p}^{m}f(z)\) reduces to the Salagean q-differential operator defined by Govindaraj and Sivasubramanian [11], and for \(p=1\), \(q\rightarrow 1\), the operator \(\mathcal{S}_{q,p}^{m}f(z)\) reduces to the Salagean differential operator defined by Salagean [26].
Taking motivation from [12] and using (1.3), we define a new class k-\(\mathcal{US}(q,\gamma ,m,p)\) of multivalent functions as follows.
Throughout paper we shall assume \(k\geq 0\), \(m\in N\cup \{0\}\), \(q\in ( 0,1 ) \), \(\gamma \in \mathbb{C} \backslash \{0\}\), and \(p\in N\).
Definition 1.1
A function \(f(z)\in \mathcal{A}_{p}\) is in the class k-\(\mathcal{US}(q,\gamma ,m,p)\) if it satisfies the condition
By taking specific values of parameters, we obtain many important subclasses studied by various authors in earlier papers. Here we enlist some of them.
-
(i)
For \(p=1\), the class k-\(\mathcal{US}(q,\gamma ,m,p)\) reduces to the class k-\(\mathcal{US}(q,\gamma ,m)\) studied by Saqib et al. [12].
-
(ii)
For \(p=1\), \(m=0\), \(k=0\), and \(\gamma \in \mathbb{C} \backslash \{0\}\), the class k-\(\mathcal{US}(q,\gamma ,m,p)\) reduces to the class \(\mathcal{S}_{q}^{\ast }(\gamma )\) studied by Seoudy and Aouf [27].
-
(iii)
For \(p=1\), \(m=0\), \(k=0\), and \(\gamma =\frac{1}{1-\alpha }\), with \(0\leq \alpha <1\), the class k-\(\mathcal{US}(q,\gamma ,m,p)\) reduces to the class \(\mathcal{S}_{q}^{\ast }(\alpha )\) studied by Agrawal and Sahoo [2].
-
(iv)
For \(p=1\), \(m=0\), \(q\rightarrow 1\), and \(\gamma =\frac{1}{1-\alpha }\), with \(0\leq \alpha <1\), the class k-\(\mathcal{US}(q,\gamma ,m,p)\) reduces to the class \(\mathcal{SD}(k,\alpha )\) studied by Shams et al. [28].
-
(v)
For \(p=1\), \(m=0\), \(q\rightarrow 1\), and \(\gamma =\frac{2}{1-\alpha }\), with \(0\leq \alpha <1\), the class k-\(\mathcal{US}(q,\gamma ,m,p)\) reduces to the class \(\mathcal{KD}(k,\alpha )\) studied by Owa et al. [24].
-
(vi)
For \(p=1\), \(k=1\), \(m=0\), \(q\rightarrow 1\), and \(\gamma =\frac{1}{1-\alpha }\), with \(0\leq \alpha <1\), the class k-\(\mathcal{US}(q,\gamma ,m,p)\) reduces to the class \(\mathcal{S}(\alpha )\) studied by Ali et al. [3].
-
(vii)
For \(p=1\), \(k=1\), \(m=0\), \(q\rightarrow 1\), and \(\gamma =\frac{2}{1-\alpha }\), with \(0\leq \alpha <1\), the class k-\(\mathcal{US}(q,\gamma ,m,p)\) reduces to the class \(\mathcal{K}(\alpha )\) studied by Ali et al. [3].
-
(viii)
For \(p=1\), \(m=0\), \(q\rightarrow 1\), the class k-\(\mathcal{US}(q,\gamma ,m,p)\) reduces to the class \(\mathcal{K}\)-\(\mathcal{ST}\) introduced by Kanas and Wisniowska [17].
-
(ix)
For \(p=1\), \(k=0\), \(m=0\), \(q\rightarrow 1\), and \(\gamma =\frac{1}{1-\alpha }\), with \(0\leq \alpha <1\), the class k-\(\mathcal{US}(q,\gamma ,m,p)\) reduces to the class \(\mathcal{S}^{\ast }(\alpha )\), a well-known class of starlike functions of order α, respectively.
Geometric interpretation. A function \(f(z)\in \mathcal{A}_{p}\) is in the class k-\(\mathcal{US}(q,\gamma ,m,p)\) if and only if \(\frac{1}{[p]_{q}} ( \frac{z\partial _{q}S_{q,p}^{m}f(z)}{S_{q,p}^{m}f(z)} ) \) takes all the values in the conic domain \(\varOmega _{k,\gamma }=h_{k,\gamma }(E)\) such that
where
Since \(h_{k,\gamma }(z)\) is convex univalent, so the above definition can be written as
where
The boundary \(\partial \varOmega _{k,\gamma }\) of the above set becomes an imaginary axis when \(k=0\), and a hyperbola when \(0< k<1\). For \(k=1\), the boundary \(\partial \varOmega _{k,\gamma }\) becomes a parabola and it is an ellipse when \(k>1\) and in this case where
and \(t\in (0,1)\) is chosen such that \(k=\cosh (\pi K^{\prime }(t)/(4K(t)))\). Here \(K(t)\) is Legendre’s complete elliptic integral of the first kind and \(K^{\prime }(t)=K(\sqrt{1-t^{2}})\), and \(K^{\prime } ( t ) \) is the complementary integral of \(K ( t ) \) (for details, see [16, 17, 23]). Moreover, \(h_{k,\gamma }(E)\) is convex univalent in E, see [16, 17]. All of these curves have the vertex at the point \(\frac{k+\gamma }{k+1}\).
2 A set of lemmas
Each of the following lemmas will be needed in our present investigation.
Lemma 2.1
([25])
Let \(h(z)=\sum_{n=1}^{\infty }h_{n}z^{n}\prec F(z)=\sum_{n=1}^{\infty }d_{n}z^{n}\) in E. If \(F(z)\) is convex univalent in E, then
Lemma 2.2
([31])
Let \(k\in {}[ 0,\infty )\) and let \(h_{k,\gamma }\) be defined (1.6). If
where \(A=\frac{2\cos ^{-1}k}{\pi }\), and \(t\in (0,1)\) is chosen such that \(k=\cosh ( \frac{\pi K^{\prime}(t)}{K(t)} ) \), \(K(t)\) is Legendre’s complete elliptic integral of the first kind.
Lemma 2.3
([18])
Let \(h(z)=1+\sum_{n=1}^{\infty }c_{n}z^{n}\) be analytic in E and satisfy \(\operatorname{Re}\{h(z)\}>0\) for z in E. Then the following sharp estimate holds:
3 Main results
In this section, we will prove our main results.
Theorem 3.1
Let \(f(z)\in k-\mathcal{US}(q,\gamma ,m,p)\). Then
where \(w(z)\) is analytic in E with \(w(0)=0\) and \(\vert w(z) \vert <1\). Moreover, for \(\vert z \vert =\rho \), we have
where \(h_{k,\gamma }(z)\) is defined by (1.6).
Proof
If \(f(z)\in k-\mathcal{US}(q,\gamma ,m,p)\), then using identity (1.5), we obtain
For some function \(w(z)\) is analytic in E with \(w(0)=0\) and \(\vert w(z) \vert <1\). Integrating (3.3) and after some simplification, we have
This proves (3.1). Noting that the univalent function \(h_{k,\gamma }(z)\) maps the disk \(|z|<\rho \) \((0<\rho \leq 1)\) onto a region which is convex and symmetric with respect to the real axis, we see
for \(z\in E\). Consequently, subordination (3.4) leads to
implies that
This completes the proof. □
When \(p=1\), we have the following known result proved by Saqib et al. in [12].
Corollary 3.2
Let \(f(z)\in k-\mathcal{US}(q,\gamma ,m)\). Then
where \(w(z)\) is analytic in E with \(w(0)=0\) and \(\vert w(z) \vert <1\). Moreover, for \(\vert z \vert =\rho \), we have
where \(h_{k,\gamma }(z)\) is defined by (1.6).
Theorem 3.3
If \(f(z)\in k-\mathcal{US}(q,\gamma ,m,p)\), then
and
where \(\delta =p \vert Q_{1} \vert \) with \(Q_{1}\) given by (2.2).
Proof
Let
where \(h(z)\) is analytic in E and \(h(0)=1\). Let \(h(z)=1+\sum _{n=1}^{\infty }c_{n}z^{n}\) and \(S_{q,p}^{m}f(z)\) be given by (1.3). Then (3.8) becomes
Now comparing the coefficients of \(z^{n+p-1}\), we obtain
Taking the absolute on both sides and then applying the coefficient estimates \(\vert c_{n} \vert \leq \vert Q_{1} \vert \), see in [23], we have
Let us take \(\delta =p \vert Q_{1} \vert \), then we have
We apply mathematical induction on (3.9), so for \(n=2\) in (3.9), we have
which shows that (3.7) holds for \(n=2\). Now consider the case \(n=3\) in (3.9), we have
Using (3.10), we have
which shows that (3.7) holds for \(n=3\). Let us assume that (3.7) is true for \(n\leq t\), that is,
Consider
which proves the assertion of theorem \(n=t+1\). Hence (3.7) holds for all n, \(n\geq 3\).
This completes the proof. □
When \(p=1\), we have the following known result proved by Saqib et al. in [12].
Corollary 3.4
([12])
If \(f(z)\in k-\mathcal{US}(q,\gamma ,m)\), then
and
where \(\delta = \vert Q_{1} \vert \) with \(Q_{1}\) given by (2.2).
Theorem 3.5
Let \(0\leq k<\infty \) be fixed and let \(f(z)\in k-\mathcal{US}(q,\gamma ,m,p)\) with the form (1.1). Then, for a complex number μ,
where
and \(\delta =p \vert Q_{1} \vert \), with \(Q_{1}\) and \(Q_{2}\) given by (2.2) and (2.3).
Proof
Let \(f(z)\in k-\mathcal{US}(q,\gamma ,m,p)\), then there exists a Schwarz function \(w(z)\), with \(w(0)=0\) and \(|w(z)|<1\), such that
Let \(h(z)\in \mathcal{P}\) be a function defined as
which gives
and
Using (3.14) in (3.13) and along with (1.3), we obtain
and
Using any complex number μ and the above coefficients, we have
Using Lemma 2.3 on (3.15), we have
where
This is our required result (3.11). □
When \(p=1\), we have the following known result proved by Saqib et al. in [12].
Corollary 3.6
([12])
Let \(0\leq k<\infty \) be fixed and let \(f(z)\in k-\mathcal{US}(q,\gamma ,m)\) with the form (1.1). Then, for a complex number μ,
where
and \(Q_{1}\) and \(Q_{2}\) are given by (2.2) and (2.3).
Theorem 3.7
If a function \(f(z)\in \mathcal{A}_{p}\) has the form (1.1) and satisfies the condition
then \(f(z)\in k-\mathcal{US}(q,\gamma ,m,p)\).
Proof
Let
From (3.16), it follows that
To show that \(f(z)\in k-\mathcal{US}(q,\gamma ,m,p)\), it is enough to prove that
From (3.17), we have
□
When \(p=1\), we have the following known result proved by Hussain et al. in [12].
Corollary 3.8
([12])
If a function \(f(z)\in \mathcal{A}\) has the form (1.1) and satisfies the condition
then \(f(z)\in k-\mathcal{US}(q,\gamma ,m)\).
When \(q\rightarrow 1\), \(p=1\), \(m=0\), \(\gamma =1-\alpha \), with \(0\leq \alpha <1\), we have the following known result, proved by Shams et al. in [28].
Corollary 3.9
A function \(f\in A\) of the form (1.1) is in the class \(\mathcal{SD}(k,\alpha )\) if it satisfies the condition
where \(0\leq \alpha <1\) and \(k\geq 0\).
When \(q\rightarrow 1\), \(p=1\), \(m=0\), \(\gamma =1-\alpha \), with \(0\leq \alpha <1\) and \(k=0\), we have the following known result proved by Silverman in [30].
Corollary 3.10
A function \(f\in A\) of the form (1.1) is in the class \(\mathcal{SD}(\alpha )\) if it satisfies the condition
Theorem 3.11
Let \(f(z)\in k-\mathcal{US}(q,\gamma ,m,p)\). Then \(f(E)\) contains an open disk of radius
where \(\delta =p \vert Q_{1} \vert \) with \(Q_{1}\) given by (2.2).
Proof
Let \(w_{0}\neq 0\) be a complex number such that \(f(z)\neq w_{0}\) for \(z\in E\). Then
Since \(f_{1}(z)\) is univalent, so
Now, by using (3.6), we have
Hence we have
□
When \(p=1\), we have the following known result proved by Saqib et al. in [12].
Corollary 3.12
([12])
Let \(f(z)\in k-\mathcal{US}(q,\gamma ,m)\). Then \(f(E)\) contains an open disk of radius
where \(Q_{1}\) is given by (2.2).
References
Adams, C.R.: On the linear partial q-difference equation of general type. Trans. Am. Math. Soc. 31, 360–371 (1929)
Agrawal, S., Sahoo, S.K.: A generalization of starlike functions of order α. Hokkaido Math. J. 46, 15–27 (2017)
Ali, R.M.: Starlikeness associated with parabolic regions. Int. J. Math. Math. Sci. 4, 561–570 (2005)
Anastassiou, G.A., Gal, S.G.: Geometric and approximation properties of generalized singular integrals in the unit disk. J. Korean Math. Soc. 43(2), 425–443 (2006)
Andrews, G.E., Askey, G.E., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)
Aral, A., Gupta, V.: On the generalized Picard and Gauss Weierstrass singular integrals. J. Comput. Anal. Appl. 8(3), 249–261 (2006)
Aral, A., Gupta, V.: On q-Baskakov type operators. Demonstr. Math. 42(1), 109–122 (2009)
Aral, A., Gupta, V.: On the Durrmeyer type modification of q-Baskakov type operators. Nonlinear Anal., Theory Methods Appl. 73(3), 1171–1180 (2010)
Aral, A., Gupta, V.: Generalized q-Baskakov operators. Math. Slovaca 61(4), 619–634 (2011)
Carmichael, R.D.: The general theory of linear q-difference equations. Am. J. Math. 34, 147–168 (1912)
Govindaraj, M., Sivasubramanian, S.: On a class of analytic functions related to conic domains involving q-calculus. Anal. Math. https://doi.org/10.1007/s10476-017-0206-5
Hussain, S., Khan, S., Zaighum, M.A., Darus, M.: Certain subclass of analytic functions related with conic domains and associated with Salagean q-differential operator. AIMS Math. 2(4), 622–634 (2017)
Jackson, F.H.: On q-functions and certain differences operator. Earth Environ. Sci. Trans. R. Soc. Edinb. 46(2), 253–281 (1909)
Jackson, F.H.: On q-definite integrals. Pure Appl. Math. Q. 41(15), 193–203 (1910)
Kanas, S., Raducanu, D.: Some class of analytic functions related to conic domains. Math. Slovaca 64(5), 1183–1196 (2014)
Kanas, S., Wisniowska, A.: Conic regions and k-uniform convexity. J. Comput. Appl. Math. 105, 327–336 (1999)
Kanas, S., Wisniowska, A.: Conic domains and k-starlike functions. Rev. Roum. Math. Pures Appl. 45, 647–657 (2000)
Ma, W., Minda, D.: A unified treatment of some special classes of univalent functions. In: Li, Z., Ren, F., Yang, L., Zhang, S. (eds.) Proceedings of the Conference on Complex Analysis, pp. 157–169. International Press, Somerville (1992)
Mahmood, S., Darus, M.: Some subordination results on q-analogue of Ruscheweyh differential operator. Abstr. Appl. Anal. 2014, Article ID 958563 (2014)
Mahmood, S., Sokol, J.: New subclass of analytic functions in conical domain associated with Ruscheweyh q-differential operator. Results Math. (2016)
Masjed-Jamei, M., Koepf, W.: Some summation theorems for generalized hypergeometric functions. Axioms 7, 38 (2018)
Mason, T.E.: On properties of the solution of linear q-difference equations with entire function coefficients. Am. J. Math. 37, 439–444 (1915)
Noor, K.I., Arif, M., Ul-Haq, W.: On k-uniformly close-to-convex functions of complex order. Appl. Math. Comput. 215, 629–635 (2009)
Owa, S., Polatoglu, Y., Yavuz, E.: Coefficient inequalities for classes of uniformly starlike and convex functions. J. Inequal. Pure Appl. Math. 7(5), 1–5 (2006)
Rogosinski, W.: On the coefficients of subordinate functions. Proc. Lond. Math. Soc. 48, 48–82 (1943)
Salagean, G.S.: Subclasses of univalent functions. In: Complex Analysis, Fifth Romanian–Finnish Seminar, Part 1, Bucharest, 1981. Lecture Notes in Mathematics, vol. 1013, pp. 362–372. Springer, Berlin (1983)
Seoudy, T.M., Aouf, M.K.: Coefficient estimates of new classes of q-starlike and q-convex functions of complex order. J. Math. Inequal. 10(1), 135–145 (2016)
Shams, S., Kulkarni, S.R., Jahangiri, J.M.: Classes of uniformly starlike and convex functions. Int. J. Math. Math. Sci. 55, 2959–2961 (2004)
Shareef, Z., Hussain, S., Darus, D.: Convolution operator in geometric functions theory. J. Inequal. Appl. 2012, 213 (2012)
Silverman, H.: Univalent functions with negative coefficients. Proc. Am. Math. Soc. 51, 109–116 (1975)
Sim, S.J., Kwon, O.S., Cho, N.E., Srivastava, H.M.: Some classes of analytic functions associated with conic regions. Taiwan. J. Math. 16(1), 387–408 (2012)
Trjitzinsky, W.J.: Analytic theory of linear q-difference equations. Acta Math. 61, 1–38 (1933)
Funding
The work here is supported by GUP-2017-064 and UKM Grant Nos. DPP-2015-FST.
Author information
Authors and Affiliations
Contributions
SH came with the main thoughts and helped to draft the manuscript, SK and MAZ proved the main theorems, MD revised the paper. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that there are no competing interests.
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
Hussain, S., Khan, S., Zaighum, M.A. et al. Applications of a q-Salagean type operator on multivalent functions. J Inequal Appl 2018, 301 (2018). https://doi.org/10.1186/s13660-018-1888-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-018-1888-3