Abstract
Examining the asymptotic growth behavior of holomorphic and meromorphic functions has significant importance in complex analysis. Estimations of upper bounds for the rate of growth of entire functions are mostly guaranteed. However, the analog estimations for lower bounds of the rate of growth are not always attainable. In this paper, we give the lower rate of growth of the generalized Hadamard product of two entire axially monogenic functions. It has also been shown that the product entire axially monogenic function is of regular growth or perfectly regular growth when its constituent entire axially monogenic functions possess these properties. The investigation of both upper and lower bounds by means of linear transmutation is also provided. Furthermore, some applications related to approximation theory are outlined.
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References
Abdalla, M., Abul-Ez, M.: The growth of generalized Hadamard product of entire axially monogenic functions. Hacet. J. Math. Stat. 47(5), 1231–1239 (2018)
Abul-Ez, M.: Hadamard product of bases of polynomials in Clifford analysis. Complex Var. 43, 109–128 (2000)
Abul-Ez, M., Abd-Elmageed, H., Hidan, M., Abdalla, M.: On the growth order and growth type of entire functions of several complex matrices. J. Funct. Spaces 2020, 1–9 (2020). https://doi.org/10.1155/2020/4027529
Abul-Ez, M., Constales, D.: Basic sets of polynomials in Clifford analysis. Complex Var. 14, 177–185 (1990)
Abul-Ez, M., Constales, D.: Linear substitution for basic sets of polynomials in Clifford analysis. Portugaliae Math 48, 143–154 (1991)
Abul-Ez, M., Constales, D.: On the order of basic series representing Clifford valued functions. Appl. Math. Comput. 142(2–3), 575–584 (2003)
Abul-Ez, M.De, Almeida, R.: On the lower order and type of entire axially monogenic functions. Results Math. 63, 1257–1275 (2013)
Bakhet, A., Jiao, Y., He, F.: On the Wright hypergeometric matrix functions and their fractional calculus. Integr. Trans. Spec. Funct. 30, 138–156 (2019)
Boas Jr., R.P.: Basic sets of polynomials (I). Duke Math. J. 15(3), 717–724 (1948)
Boas, R.P., Buck, R.C.: Polynomial Expansions of Analytic Functions. Springer, Berlin (1985)
Bock, S., Gürlebeck, K.: On a generalized Appell system and monogenic power series. Math. Methods Appl. Sci. 33, 394–411 (2010)
Cação, I., Gürlebeck, K., Malonek, H.: Special monogenic polynomials and \(L_2\)-approximation. Adv. Appl. Clifford Algebras 11, 47–60 (2001)
Choi, J.H., Kim, Y.C., Owa, S.: Generalizations of Hadamard products of functions with negative coefficients. J. Math. Anal. Appl. 199, 495–501 (1996)
Constales, D., De Almeida, R., Kraußhar, R.: On the relation between the growth and the Taylor coefficients of entire solutions to the higher dimensional Cauchy–Riemann system in \({\mathbb{R}}^{n+1}\). J. Math. Anal. Appl. 327, 763–775 (2007)
Constales, D., De Almeida, R., Kraußhar, R.: On the growth type of entire monogenic functions. Arch. Math. 88, 153–163 (2007)
Constales, D., De Almeida, R., Kraußhar, R.: Applications of the maximum term and the central index in the asymptotic growth analysis of entire solutions to higher dimensional polynomial Cauchy–Riemann equations. Complex Var. Elliptic Equ. 53, 195–213 (2008)
De Almeida, R., Kraußhar, R.: On the asymptotic growth of entire monogenic functions. Z. Anwend. 24(4), 791–813 (2005)
De Almeida, R., Kraußhar, R.: Basics on growth orders of polymonogenic functions. Complex Var. Elliptic Equ. 60, 1–25 (2015)
Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis. Research Notes in Mathematics 76. London Pitman Books Ltd, London (1982)
Dutta, R.K.: On order of a function of several complex variables analytic in the unit polydisc. Krag. J. Math. 36, 163–174 (2012)
Goldberg, A.A.: Elementary remarks on the formulas defining order and type of functions of several variables (Russian), Doklady Akademii Nauk Armyanskii, SSR29, 145–151 (1959)
Gürlebeck, K., Habetha, K., Sprössig, W.: Holomorphic functions in the plane and \(n\)-dimensional space. Birkh aü ser- Verlag, Basel (2008)
Gürlebeck, K., Sprössig, W.: Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, Chichester (1997)
Gürlebeck, K., Habetha, K., Sprössig, W.: Application of Holomorphic Functions in Two and Higher Dimensions. Springer, Basel (2016)
Hayman, W.K.: The local growth of power series: a survey of the Wiman–Valiron method. Canadian Mathematical Bulletin 17, 317–358 (1974)
Jank, G., Volkmann, L.: Meromorphe Funktionen und Differentialgleichungen. UTB Birkhäuser, Basel (1985)
Kishka, Z., Abul-Ez, M., Saleem, M., Abd-Elmaged, H.: On the order and type of entire matrix functions in complete Reinhardt domain. J. Mod. Meth. Numer. Math. 3, 31–40 (2012)
Kumar, S., Bala, K.: Generalized type of entire monogenic functions of slow growth. Transylv. J. Math. Mech. 3(2), 95–102 (2011)
Kumar, S., Bala, K.: Generalized order of entire monogenic functions of slow growth. J. Nonlinear Sci. Appl. 5(6), 418–425 (2012)
Kumar, S., Bala, K.: Generalized growth of monogenic Taylor series of finite convergence radius. Annali delĺ Universitá di Ferrara VII: Scienze Matematiche 59(1), 127–140 (2013)
Lindelöf, E.: Sur la détermination de la croissance des fonctions entières définies par un développement de Taylor. Darb. Bull. 27(2), 213–226 (1903)
Malonek, H.: Power series representation for monogenic functions in \({\mathbb{R}}^{m+1}\) based on a permutational product. Complex Var. 15, 181–191 (1990)
Nevanlinna, R.: Zur Theorie der meromorphen Funktionen. Acta Math. 46, 1–99 (1925)
Pringsheim, A.: Elementare Theorie der ganzen transzendenten Funktionen von endlicher Ordnung. Math. Ann. 58, 257–342 (1904)
Ronkin, L.I.: Introduction to the theory of entire functions of several variables. Translations of Mathematical Monographs, vol. 44. American Mathematical Society, VI, Providence RI (1974)
Saleem, M., Abul-Ez, M., Zayed, M.: On polynomial series expansions of Cliffordian functions. Math Methods Appl. Sci. 35, 134–143 (2012)
Shang, Y.: Lower bounds for Gaussian Estrada index of graphs. Symmetry 10(8), 325 (2018)
Simon, M., Suslov, S.: Expansion of analytic functions in q-orthogonal polynomials. Ramanujan J. 19(3), 281–303 (2009)
Srivastava, G.S.: A note on growth of generalized Hadamard product of entire functions. Southeast Asian Bull. Math. 33(1), 147–152 (2009)
Srivastava, R.K., Kumar, V.: On the order and type of integral functions of several complex variables. Compos. Math. 17, 161–166 (1966)
Srivastava, G.S., Kumar, S.: On the generalized order and generalized type of entire monogenic functions. Demonstr. Math. 46(4), 663–677 (2013)
Sprössig, W.: Clifford analysis and its applications in mathematical physics. Cubo Matemáitica Educacional 4 2, 253–314 (2002)
Valiron, G.: Lectures on the General Theory of Integral Functions. Chelsea, New York (1949)
Wiman, A.: Über den Zusammenhang zwischen dem Maximalbetrage einer analytischen Funktion und dem gröten Gliede der zugehörigen Taylorschen Reihe. Acta Math. 37, 305–326 (1914)
Whittaker, J., Gattegno, C.: Sur les séries de base de polynbmes quelconques. Gauthier-Villars, Paris (1949)
Zayed, M.: Generalized Hadamard product bases of special monogenic polynomials. Adv. Appl. Clifford Algebras 30, 10 (2020)
Acknowledgements
The author gratefully appreciates the kind support provided by the Deanship of Scientific Research at King Khalid University, Saudi Arabia, under Grant GRP-82-41.
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Communicated by V. Ravichandran.
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Zayed, M. Lower Growth of Generalized Hadamard Product Functions in Clifford Setting. Bull. Malays. Math. Sci. Soc. 44, 805–826 (2021). https://doi.org/10.1007/s40840-020-00983-y
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DOI: https://doi.org/10.1007/s40840-020-00983-y
Keywords
- Clifford algebra
- Axially monogenic function
- Growth order
- Growth type
- Lower order and type
- Hadamard product