In the paper, a new class of Bazilevič harmonic functions is introduced. First, the sufficient and necessary conditions and integral expressions of the class are proved by the subordination relationship and the basic theory of harmonic functions. Then, the inclusion relation and radius problems are explored, yielding intriguing new findings.
Citation: Shuhai Li, Lina Ma, Xiaomeng Niu, Huo Tang. Some properties of Bazilevič harmonic functions[J]. AIMS Mathematics, 2025, 10(8): 18824-18837. doi: 10.3934/math.2025841
In the paper, a new class of Bazilevič harmonic functions is introduced. First, the sufficient and necessary conditions and integral expressions of the class are proved by the subordination relationship and the basic theory of harmonic functions. Then, the inclusion relation and radius problems are explored, yielding intriguing new findings.
| [1] | P. L. Duren, Univalent functions, New York: Springer-Verlag, 1983. |
| [2] | I. E. Bazilevič, On a case of integrability in quadratures of the Loewner-Kufarev equation, Mat. Sb., 79 (1955), 471–476. |
| [3] |
R. Singh, On Bazilevič functions, P. Am. Math. Soc., 38 (1973), 261–271. http://dx.doi.org/10.1090/S0002-9939-1973-0311887-9 doi: 10.1090/S0002-9939-1973-0311887-9
|
| [4] | M. S. Liu, The radius of univalence for certain class of analytic functions, In: Boundary Value Problems, Integral Equations and Related Problems, Singapore: World Scientific Publishing, 2000,122–128. |
| [5] |
J. Clunie, T. S. Small, Harmonic univalent functions, Fenn. Math., 9 (1984), 3–25. https://doi.org/10.5186/aasfm.1984.0905 doi: 10.5186/aasfm.1984.0905
|
| [6] | P. Duren, Harmonic mappings in the plane, Cambridge: Cambridge University Press, 2004. https://doi.org/10.1017/CBO9780511546600 |
| [7] |
B. K. Chinhara, P. Gochhayat, S. Maharana, On certain harmonic mappings with some fixed coefficients, Monatsh. Math., 190 (2019), 261–280. https://doi.org/10.1007/s00605-018-1228-1 doi: 10.1007/s00605-018-1228-1
|
| [8] | D. Klimek-Smȩt, A. Michalski, Univalent anti-analytic perturbations of convex analytic mappings in the unit disc, Ann. Univ. Mariae Curie-Skłodowsk, 61 (2007), 39–49. |
| [9] |
S. S. Miller, P. T. Mocanu, Differential subordination and univalent functions, Mich. Math. J., 28 (1981), 157–171. https://doi.org/10.1307/mmj/1029002507 doi: 10.1307/mmj/1029002507
|
| [10] | F. G. Avkhadiev, K. J. Wirths, Schwarz-Pick type inequalities, Frontiers in Mathematics, Basel, Boston, Berlin: Birkhauser Verlag AG, 2009. https://doi.org/10.1007/978-3-0346-0000-2 |
| [11] |
S. D. Bernardi, New distortion theorems for functions of positive real part and applications to partial sums of univalent convex functions, P. Am. Math. Soc., 45 (1974), 113–118. https://doi.org/10.1090/s0002-9939-1974-0357755-9 doi: 10.1090/s0002-9939-1974-0357755-9
|
| [12] | W. C. Ma, On $\alpha$-convex functions of order $\beta$, Acta Math. Sin., 29 (1986), 207–212. |
| [13] |
S. Kanas, S. Maharana, J. K. Prajapat, Norm of the pre-Schwarzian derivative, Blochs constant and coefficient bounds in some classes of harmonic mappings, J. Math. Anal. Appl., 474 (2019), 931–943. https://doi.org/10.1016/j.jmaa.2019.01.080 doi: 10.1016/j.jmaa.2019.01.080
|
| [14] |
S. I. Kanas, D. Klimek-Smet, Harmonic mappings related to functions with bounded boundary rotation and norm of the pre-Schwarzian derivative, B. Korean Math. Soc., 51 (2014), 803–812. https://doi.org/10.4134/bkms.2014.51.3.803 doi: 10.4134/bkms.2014.51.3.803
|
| [15] | Y. Polatoǧlu, A. Şen, Some results on subclasses of Janowski $\lambda$-spirallike functions of complex order, Gen. Math., 51 (2007), 88–97. |
Figures(2)
Shuhai Li, Lina Ma, Xiaomeng Niu, Huo Tang. Some properties of Bazilevič harmonic functions[J]. AIMS Mathematics, 2025, 10(8): 18824-18837. doi: 10.3934/math.2025841
DownLoad: