On analytic multivalent functions associated with lemniscate of Bernoulli

Qaiser Khan; Muhammad Arif; Bakhtiar Ahmad; Huo Tang; Qaiser Khan; Muhammad Arif; Bakhtiar Ahmad; Huo Tang

doi:10.3934/math.2020149

AIMS Mathematics

2020, Volume 5, Issue 3: 2261-2271. doi: 10.3934/math.2020149

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Research article

On analytic multivalent functions associated with lemniscate of Bernoulli

1.
School of Mathematics and Statistics, Chifeng University, Chifeng 024000, Inner Mongolia, People’s Republic of China
2.
Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan

Received: 25 October 2019 Accepted: 07 February 2020 Published: 02 March 2020
MSC : 30C45, 30C50

In this paper, we establish some sufficient conditions for analytic functions associated with lemniscate of Bernoulli. In particular, we determine conditions on $\alpha $ such that $ \begin{equation*} 1+\alpha \frac{z^{2+p\left( j-1\right) }g^{\prime }\left( z\right) }{ pg^{j}\left( z\right) },\text{ for each }j = 0,1,2,3, \end{equation*} $ are subordinated by Janowski function, then $\frac{g\left(z\right) }{z^{p}} \prec \sqrt{1+z}, \ \left(z\in \mathfrak{D}\right) $. By choosing particular values of functions $g, $ we obtain some sufficient conditions for multivalent starlike functions associated with lemniscate of Bernoulli.
- multivalent functions,
- subordination,
- lemniscate of Bernoulli,
- Janowski functions
Citation: Qaiser Khan, Muhammad Arif, Bakhtiar Ahmad, Huo Tang. On analytic multivalent functions associated with lemniscate of Bernoulli[J]. AIMS Mathematics, 2020, 5(3): 2261-2271. doi: 10.3934/math.2020149

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Abstract

In this paper, we establish some sufficient conditions for analytic functions associated with lemniscate of Bernoulli. In particular, we determine conditions on $\alpha $ such that $ \begin{equation*} 1+\alpha \frac{z^{2+p\left( j-1\right) }g^{\prime }\left( z\right) }{ pg^{j}\left( z\right) },\text{ for each }j = 0,1,2,3, \end{equation*} $ are subordinated by Janowski function, then $\frac{g\left(z\right) }{z^{p}} \prec \sqrt{1+z}, \ \left(z\in \mathfrak{D}\right) $. By choosing particular values of functions $g, $ we obtain some sufficient conditions for multivalent starlike functions associated with lemniscate of Bernoulli.

References

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