Korean J. Math. Vol. 31 No. 2 (2023) pp.229-242
DOI: https://doi.org/10.11568/kjm.2023.31.2.229

Order, type and zeros of analytic and meromorphic functions of $\left[p,q\right]-\phi$ order in the unit disc

Main Article Content

Pulak Sahoo
Nityagopal Biswas


In this paper, we investigate the $\left[p,q\right]-\phi$ order and $\left[p,q\right]-\phi$ type of $f_1+f_2$, $f_1f_2$ and $\frac{f_1}{f_2}$, where $f_1$ and $f_2$ are analytic or meromorphic functions with the same $\left[p,q\right]-\phi$ order and different $\left[p,q\right]-\phi$ type in the unit disc. Also, we study the $\left[p,q\right]-\phi$ order and $\left[p,q\right]-\phi$ type of different $f$ and its derivative. At the end, we investigate the relationship between two different $\left[p,q\right]-\phi$ convergence exponents of $f$. We extend some earlier precedent well known results.

Article Details


[1] S. Bank, General theorem concerning the growth of solutions of first-order algebraic differential euqations, Composition Math. 25 (1972), 61–70. Google Scholar

[2] B. Belaidi, Growth of solutions to linear equations with analytic coefficients of [p,q]-order in the unit disc, Electron. J. Diff. Equ. 156 (2) (2011), 25–38. Google Scholar

[3] T.B. Cao, and H.X. Yi, The growth of solutions of linear equations with coefficients of iterated order in the unit disc, J. Math. Anl. Appl. 319 (2006), 278–294. Google Scholar

[4] I. Chyzhykov, J. Heittokangas, J. Rattya, Finiteness of φ− order of solutions of linear differential equations in the unit disc, J.Anal.Math. 109 (2009), 163-198. Google Scholar

[5] W. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964. Google Scholar

[6] W. Hayman, On the characteristic of functions meromorphic in the unit disc and of their integrals, Acta Math., 112 (1964), 181-214. Google Scholar

[7] J. Heittokangas, On complex differential equations in the unit disc, Ann. Acad. Sci. Fenn. Math. Diss., 122 (2000), 1-54. Google Scholar

[8] J. Heittokangas,R. Korhonen. and J. Rattya, Fast growing solutions of linear differential equations in the unit disc, Results Math., 49 (2006), 265-278. Google Scholar

[9] O.P. Juneja, G.P. Kapoor and S.K. Bajpai, On the (p,q)-order and lower (p,q)-order of an entire function, J. Reine Angew. Math., 282 (1976), 53-67. Google Scholar

[10] O.P. Juneja, G.P. Kapoor and S.K. Bajpai, On the (p,q)-type and lower (p,q)-type of an entire function, J. Reine Angew. Math., 290 (1977), 180-190. Google Scholar

[11] I. Lain, Nevanlinna Theory and Complex Differential Equations, de Gruyter, Berlin, 1993. Google Scholar

[12] B.Ja. Levin, Distribution of Zeros of Entire Functions, revised edition Transl. Math. Monographs, Vol. 5, Amer. Math. Soc. Providence, 1980. Google Scholar

[13] L.M. Li and T.B. Cao, Solutions for differential equations with moromorphic coefficients of (p,q)-order in the plane, Electron. J. Diff. Equ., 2012 (195) (2012), 1–15. Google Scholar

[14] C. Linden, The representation of regular functions, J.London Math. Soc., 39 (1964), 19–30. Google Scholar

[15] J. Lu, J. Tu and L.Z. Shi, Linear differential equations with coefficients of (p,q)-order in the complex plane, J.Math.Anal.Appl, 372 (2010), 55–67. Google Scholar

[16] X. Shen, J. Tu and H. Y. Xu, Complex oscillation of a second-order linear differential equation with entire coefficients of [p,q]–φ order, Advances in Difference Equations 2014 (2014), 2014:200. Google Scholar

[17] D. Shea, L.R. Sons. Value distribution theory for meromorphic functions of slow growth in the disc, Houston J. Math. 12 (2) (1986), 249–266. Google Scholar

[18] L.R. Sons, Unbounded functions in the unit disc, Internet. J. Math. Math. Sci., 6 (2) (1983), 201–242. https://doi.org/10.1155/S0161171283000204 Google Scholar

[19] M. Tsuji, Potential Theory in Modern Function Theory, Chelsea, Nwe York, 1975, reprint of the 1959 edition. Google Scholar

[20] J. Tu, Y. Zeng, H.Y.Xu, The Order and Type of Meromorphic Functions and Entire Functions of Finite Iterated Order, J. Computational Analysis and Applications, 21 (2016), no.5, 994–1003. Google Scholar

[21] L. Yang, Value Distribution Theory and Its New Research, Science Press, Beijing, 1982 (in Chinese). Google Scholar

[22] H.X. Yi and C.C. Yang, The Uniqueness Theory of Meromorphic Function, Science Press, Beijing, 1995(in Chinese). Google Scholar