Abstract
We introduce a new class of harmonic function $f$, that is subclass of planar harmonic mapping associated with $q-$difference operator. Let $h$ and $g$ are analytic functions in the open unit disc $\mathbb{D}=\{ z\,:\,|z|<1 \}$. If $f=h+\bar{g}$ is the solution of the non-linear partial differential equation $w_q(z)=\dfrac{D_q g(z)}{D_q h(z)}=\dfrac{\bar{f}_\bar{z}}{f_z}$ with $|w_q(z)|<1$, $w_q(z)\prec b_1 \dfrac{1+z}{1-qz}$ and $h$ is $q-$convex function of complex order, then the class of such functions are called $q-$harmonic functions for which analytic part is $q-$convex functions of complex order denoted by $\mathcal{S}_{ \mathcal{H}\mathcal{C}_q(b)}$. Obviously that the class $\mathcal{S}_{ \mathcal{H}\mathcal{C}_q(b)}$ is the subclass of $\mathcal{S}_\mathcal{H}$. In this paper, we investigate properties of the class $\mathcal{S}_{ \mathcal{H}\mathcal{C}_q(b)}$ by using subordination techniques.
References
- Andrews, G.E. Applications of basic hypergeometric functions, SIAM Rev. 16 (1974), 441- 484.
- Çetinkaya, A. and Mert, O. A certain class of harmonic mappings related to functions of bounded boundary rotation, Proc. of 12th Symposium on Geometric Function Theory and Applications (2016), 67-76.
- Duren, P. Harmonic mappings in the plane, Cambridge Tracts in Math. 2004.
- Fine, N.J. Basic hypergeometric series and applications, Math. Surveys Monogr. 1988.
- Gasper, G. and Rahman, M. Basic hypergeometric series, Cambridge University Press, 2004.
- Goodman, A.W. Univalent functions Volume I and II, Polygonal Pub. House, 1983.
- Jack, I.S. Functions starlike and convex of order $\alpha$, J. Lond. Math. Soc. (2), 3 (1971), 469-474.
- Jackson, F.H. On $q-$functions and a certain difference operator, Trans. Roy. Soc. Edin. 46 (1908), 253-281.
- Jackson, F.H. On $q-$difference integrals, Quart. J. Pure Appl. Math. 41 (1910), 193-203.
- Kac, V. and Cheung, P. Quantum calculus, Springer, 2001.
- Lewy, H. On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc. 42 (1936), 689-692.
- Polatoglu, Y., Aydogan, M. and Mert, O. Some properties of $q-$convex functions of complex order, Sarajevo J. Math. Submitted, 2016.
Abstract
References
- Andrews, G.E. Applications of basic hypergeometric functions, SIAM Rev. 16 (1974), 441- 484.
- Çetinkaya, A. and Mert, O. A certain class of harmonic mappings related to functions of bounded boundary rotation, Proc. of 12th Symposium on Geometric Function Theory and Applications (2016), 67-76.
- Duren, P. Harmonic mappings in the plane, Cambridge Tracts in Math. 2004.
- Fine, N.J. Basic hypergeometric series and applications, Math. Surveys Monogr. 1988.
- Gasper, G. and Rahman, M. Basic hypergeometric series, Cambridge University Press, 2004.
- Goodman, A.W. Univalent functions Volume I and II, Polygonal Pub. House, 1983.
- Jack, I.S. Functions starlike and convex of order $\alpha$, J. Lond. Math. Soc. (2), 3 (1971), 469-474.
- Jackson, F.H. On $q-$functions and a certain difference operator, Trans. Roy. Soc. Edin. 46 (1908), 253-281.
- Jackson, F.H. On $q-$difference integrals, Quart. J. Pure Appl. Math. 41 (1910), 193-203.
- Kac, V. and Cheung, P. Quantum calculus, Springer, 2001.
- Lewy, H. On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc. 42 (1936), 689-692.
- Polatoglu, Y., Aydogan, M. and Mert, O. Some properties of $q-$convex functions of complex order, Sarajevo J. Math. Submitted, 2016.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Mathematics |
| Authors | |
| Publication Date | August 1, 2018 |
| Published in Issue | Year 2018 Volume: 47 Issue: 4 |