Korean J. Math. Vol. 30 No. 3 (2022) pp.467-474
DOI: https://doi.org/10.11568/kjm.2022.30.3.467

Boundedness of $\mathcal{C}^{b,c}$ operators on Bloch spaces

Main Article Content

Pankaj Kumar Nath
Sunanda Naik


In this article, we consider the integral operator $\mathcal{C}^{b,c}$, which is
defined as follows:
$$ \mathcal{C}^{b,c}(f)(z)=\int_0^z \frac{f(w)*F(1,1;c;w)}{w(1-w)^{b+1-c}}dw, $$
where $*$ denotes the Hadamard/ convolution product of power series, $F(a,b;c;z)$ is the classical
hypergeometric function with $b,c>0, b+1>c$ and $f(0)=0$.
We investigate the boundedness of the $\mathcal{C}^{b,c}$ operators on Bloch spaces.

Article Details

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[1] G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge Univ. Press, (1999). Google Scholar

[2] M. R. Agrawal, P. G. Howlett, S. K. Lucas, S. Naik and S. Ponnusamy,Boundedness of generalized Cesaro averaging operators on certain function spaces, J. Comput. Appl. Math. 180 (2005), 333–344 Google Scholar

[3] R. Balasubramanian, S. Ponnusamy and M. Vuorinen, On hypergeometric functions and function spaces, J. Comput. Appl. Math. 139 (2) (2002), 299–322. Google Scholar

[4] H. Deng, S. Ponnusamy, and J. Qiao, Extreme points and support points of families of harmonic Bloch mappings, Potential Analysis 55 (2021), 619–638. Google Scholar

[5] E. Diamantopoulos and A. G. Siskakis, Composition operators and the Hilbert matrix, Studia Math. 140 (2000), 191–198. Google Scholar

[6] H. Hidetaka, Bloch-type spaces and extended Ces ́aro operators in the unit ball of a complex Banach space, Preprint (https://arxiv.org/abs/1710.11347). Google Scholar

[7] S. Kumar and S. K. Sahoo, Properties of β-Ces ́aro operators on α-Bloch Space, Rocky Mountain J. Math. 50 (1) (2020), 1723–1743. Google Scholar

[8] G. Liu and S. Ponnusamy, On Harmonic ν-Bloch and ν-Bloch-type mappings, Results in Mathematics 73(3) (2018), Art 90, 21 pages. Google Scholar

[9] J. Miao, The Ces ́aro operator is bounded on Hp for 0 < p < 1,Proc. Amer. Math. Soc. 116 (4) (1992), 1077–1079. Google Scholar

[10] M. Huang, S. Ponnusamy, and J. Qiao, Extreme points and support points of harmonic alpha-Bloch Mappings, Rocky Mountain J. Math. 50 (4) (2020), 1323–1354. Google Scholar

[11] S. Ponnusamy and M. Vuorinen, Asymptotic expansions and inequalities for hypergeometric functions, Mathematika 44 (1997), 278–301. Google Scholar

[12] A. Siskakis, Semigroups of composition operators in Bergman spaces, Bull. Austral.Math. Soc. 35 (1987), 397–406. Google Scholar

[13] A. G. Siskakis, The Cesa ́ro operator is bounded on H1, Proc. Amer. Math. Soc. 110 (4) (1990), 461–462. Google Scholar

[14] S. Stevi ́c, Boundedness and Compactness of an integral operator on mixed norm spaces on the polydisc, Sibirsk. Math. Zh. 48 (3) (2007), 694–706. Google Scholar

[15] J. Xiao, Cesa ́ro type operators on Hardy, BMOA and Bloch spaces, Arch. Math. 68 (1997), 398–406. Google Scholar

[16] K. Zhu, Operator Theory in Function Spaces, Second Edition, Math. Surveys and Monographs, 138 (2007). Google Scholar

[17] K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Springer, USA, (2005). Google Scholar