Korean J. Math. Vol. 24 No. 3 (2016) pp.369-374
DOI: https://doi.org/10.11568/kjm.2016.24.3.369

Sharp hereditary convex radius of convex harmonic mappings under an integral operator

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Published: Sep 30, 2016
Keywords:
Harmonic mapping, Convex radius, Integral operator, DCP
Mathematics Subject Classification:
30C99, 30C62

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Xingdi Chen
Department of Mathematics Huaqiao University Quanzhou, Fujian, 362021, P. R. China
Jingjing Mu
Department of Mathematics Huaqiao University Quanzhou, Fujian, 362021, P. R. China

Abstract

In this paper, we study the hereditary convex radius of convex harmonic mapping $f(z)=f_1(z)+\overline{f_2(z)}$ under the integral operator $I_{f}(z)=\int_{0}^{z}\frac{f_1(u)}{u}du+\overline{\int_{0}^z\frac{f_2(u)}{u}}$ and obtain the sharp constant $\frac{4\sqrt{6}-\sqrt{15}}{9}$, which generalized the result corresponding to the class of analytic functions given by Nash.


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Supporting Agencies

This work was supported by NNSF of China (11471128) the Natural Science Foundation of Fujian Province of China (2014J01013) NCETFJ Fund (2012FJ-NCET-ZR05) Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao

References

[1] J.W. Alexander, Functions which map the interior of the unit circle upon simple regions, Ann. of Math. 17 (1915), 12–22. Google Scholar

[2] St. Ruscheweyh and L.C. Salinas, On the preservation of direction-convexity and the Goodman-Saff conjecture, Ann. Acad. Sci. Fenn. Ser. AI Math. 14 (1989), no. 1, 63–73. Google Scholar

[3] F. Rønning, On the preservation of direction convexity under differentiation and integration, Rocky Mountain J. Math. 34 (2004), no. 3, 1121–1130. Google Scholar

[4] C.M. Pokhrel, St. Ruscheweyh and G.B. Thapa, The DCP radius of the expo- nential function, Complex Var. Theory Appl. 50 (2005), no. 7-11, 645–651. Google Scholar

[5] S. Nagpal and V. Ravichandran, Fully starlike and fully convex harmonic mappings of order α, Ann. Polon. Math. 108 (2013), no. 1, 85–107. Google Scholar

[6] S. Nagpal and V. Ravichandran,Construction of subclasses of univalent harmonic mappings, J. Korean Math. Soc. 51 (2014), no. 3, 567–592. Google Scholar

[7] A.W. Goodman and E.B. Saff, On univalent functions convex in one direction, Proc. Amer. Math. Soc. 73 (1979), no. 2 , 183–187. Google Scholar

[8] E.M. Nash, Unterschungen u ̈ber DCP-Funktionen, Diplomarbeit Bay rishes Julius-Maximilians-Universit ̈at Wu ̈rzburg, 1995. Google Scholar