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On a subclass of quasi-convex functions with fixed second coefficient

G. Thirupathi

R.M.K. Engineering College, R.S.M. Nagar, Kavaraipettai-601 206, Chennai, Tamil Nadu, India

https://doi.org/10.1142/S1793557117500784Cited by:2 (Source: Crossref)

Abstract

In this paper, we introduce the concept of quasi-convex functions by considering a subclass $𝒞_{p}^{*} (λ, β)$ of normalized analytic functions in the unit disk $𝕌$ for some convex function $g (z)$ with fixed second coefficient of order $λ$ , such that

Re{(zf′(z))′g′(z)}≥β.<math display="block" altimg="eq-00005.gif"><mrow><mstyle><mtext class="textrm" mathvariant="normal">Re</mtext></mstyle><mfenced separators="" open="{" close="}"><mrow><mfrac><mrow><msup><mrow><mo class="MathClass-open" stretchy="false">(</mo><mi>z</mi><msup><mrow><mi>f</mi></mrow><mrow><mi>′</mi></mrow></msup><mo class="MathClass-open" stretchy="false">(</mo><mi>z</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-close" stretchy="false">)</mo></mrow><mrow><mi>′</mi></mrow></msup></mrow><mrow><msup><mrow><mi>g</mi></mrow><mrow><mi>′</mi></mrow></msup><mo class="MathClass-open" stretchy="false">(</mo><mi>z</mi><mo class="MathClass-close" stretchy="false">)</mo></mrow></mfrac></mrow></mfenced><mo class="MathClass-rel">≥</mo><mi>β</mi><mo class="MathClass-punc">.</mo></mrow></math>

We obtain results on growth and distortion theorems and radius of convexity.

Communicated by A. Escassut

Keywords:

AMSC: 30C45

References

1. H. S. Al-Amiri, On close-to-star functions of order $α$ , Proc. Amer. Math. Soc. 29 (1971) 103–108. Web of Science, Google Scholar
2. M. Finkelstein, Growth estimates of convex functions, Proc. Amer. Math. Soc. 18 (1967) 412–418. Crossref, Web of Science, Google Scholar
3. X. Liangpeng and L. Xiaoli, A general subclass of close-to-convex functions, Kragujevac J. Math. 36(2) (2012) 251–260. Google Scholar
4. R. J. Libera, Some radius of convexity problems, Duke Math. J. 31 (1964) 143–158. Crossref, Web of Science, Google Scholar
5. K. I. Noor, Some classes of alpha-quasi-convex functions, Int. J. Math. Math. Sci. 11(3) (1988) 497–501. Crossref, Google Scholar
6. Z. Peng, On a subclass of close to convex functions, Acta Math. Sci. Ser. B Engl. Ed. 30(5) (2010) 1449–1456. Google Scholar
7. M. S. Robertson, On the theory of univalent functions, Ann. Math. 37 (1936) 374–408. Crossref, Google Scholar
8. C. Selvaraj, A subclass of close-to-convex functions, Southeast Asian Bull. Math. 28(1) (2004) 113–123. Google Scholar
9. C. Selvaraj, S. Stelin and S. Logu, On a subclass of close-to-convex functions associated with fixed second coefficient, Appl. Comput. Math. 4(5) (2015) 342–345. Crossref, Google Scholar
10. C. Selvaraj and G. Thirupathi, Certain generalization of the Noor functions, Int. J. Math. Anal. 9(17) (2015) 847–855. Crossref, Google Scholar
11. H. Silverman, On a close-to-convex functions, Proc. Amer. Math. Soc. 36(2) (1972) 477–484. Crossref, Web of Science, Google Scholar