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\begin{document}

\sloppy
\noindent
{\bf  {\textquotedblleft Vasile Alecsandri" University  of Bac\u au \\
Faculty of Sciences \\
Scientific Studies and Research \\
Series Mathematics and Informatics \\
Vol. 35 (2025), No. 1, 5 - 10} }

\vspace{0.5cm}

\title[Short title]{Long title  $(\alpha ,\beta )-$order}
\author[Short names authors ]{Authors}
\maketitle

\vspace{0.5cm}

\textbf{Abstract. }In this paper, we wish to investigate the ... and obtain some results which improve and generalize
some previous results of Tu et al. \cite{t1} as well as the results from \cite%
{b1, b2, b3}.

\section{\textbf{Introduction}}

\qquad Throughout this paper, we assume that the reader is familiar with the
fundamental results and the standard notations of the Nevanlinna value
distribution theory of entire and meromorphic functions and the theory of
complex linear differential equations which are available in ...
and therefore we do not explain those in details. 


\vspace{1cm}

-------------------------------------- \newline
\textbf{Keywords and phrases:}  Differential equations, $(\alpha ,\beta )$-order, growth of
solutions.\newline
\textbf{(2020) Mathematics Subject Classification:} 30D35, 34M10. \newpage
\begin{equation}
f^{(k)}(z)+A_{k-1}(z)f^{(k-1)}(z)+\cdots +A_{0}(z)f(z)=0  \label{1}
\end{equation}%
and%
\begin{equation}
f^{(k)}(z)+A_{k-1}(z)f^{(k-1)}(z)+\cdots +A_{0}(z)f(z)=F(z)  \label{2}
\end{equation}%
and achieved many valuable results when the coefficients $A_{0}(z),...,$ $ A_{k-1}(z),$ $F(z)$ $(k\geq 2)$ in (\ref{1}) or (\ref{2}) are
entire or meromorphic functions of finite order or finite iterated $p$-order
or $(p,q)$-th order or $(p,q)$-$\varphi $ order (e.g. \cite{4}, \cite{b},
\cite{bou}, \cite{g}, \cite{20}, \cite{13}, \cite{10}-\cite{l}, \cite{STX},
\cite{t1}-\cite{t3}, \cite{15}).




\qquad In this paper, our aim is to make use of the concepts of entire
functions of $(\alpha ,\beta )$-order or generalized order after giving a
minor modification to the original definition (e.g. see, \cite{MST, MNS}) in
order to investigate the complex linear differential equations (\ref{1}) or (%
\ref{2}).

\section{\textbf{Definitions and Notations}}

\qquad First of all, let $L$ be a class of continuous non-negative on $%
(-\infty ,+\infty )$ function $\alpha $ such that 

\begin{proposition}
\label{p1} If $f(z)$ is an entire function, then%
\begin{equation*}
\sigma _{(\alpha (\log ),\beta )}[f]=\underset{r\rightarrow +\infty }{\lim
\sup }\frac{\alpha (\log ^{[2]}T(r,f))}{\beta (\log r)}=\underset{%
r\rightarrow +\infty }{\lim \sup }\frac{\alpha (\log ^{[3]}M(r,f))}{\beta
(\log r)}\text{.}
\end{equation*}
\end{proposition}

\begin{proof}
By the inequality $T(r,f)\leq \log ^{+}M(r,f)\leq \frac{R+r}{R-r}T(R,f)$ ($%
0<r<R$) (cf. \cite{1}) for an entire function $f$, set $R=2r,$ we have
\begin{equation*}
T(r,f)\leq \log ^{+}M(r,f)\leq 3T(2r,f)\text{.}
\end{equation*}%
By using $\alpha ((1+o(1))x)=(1+o(1))\alpha (x)$ and $\beta
((1+o(1))x)=(1+o(1))\beta (x)$ as $x\rightarrow +\infty $, one can easily
obtain the proposition.
\end{proof}

\begin{definition}
\label{d2} If $f(z)$ is an entire function satisfying $0<\sigma _{(\alpha
,\beta )}[f]<+\infty $, then for any $\gamma \in L$ and $\gamma (r)\neq r$,%
\begin{equation*}
\left\{
\begin{array}{l}
\text{ }\sigma _{(\gamma (\alpha ),\beta )}[f]=+\infty ~\text{\ \ when }%
\gamma (\alpha )\in L_{si}\text{ and }\underset{r\rightarrow +\infty }{\lim }%
\frac{\gamma (\sigma \beta (\log r))}{\beta (\log r)}=+\infty\\
\text{ for any } \sigma <\sigma _{(\alpha ,\beta )}[f]\text{;} \\
\sigma _{(\gamma (\alpha ),\beta )}[f]=0~\text{\ \ when }\gamma (\alpha )\in
L_{si}\text{ and }\underset{r\rightarrow +\infty }{\lim }\frac{\gamma
(\sigma _{1}\beta (\log r))}{\beta (\log r)}=0\\
\text{ for any }\sigma_{1}>\sigma _{(\alpha ,\beta )}[f]\text{;} \\
\sigma _{(\alpha ,\gamma (\beta ))}[f]=+\infty ~\ \ \text{when }\gamma
(\beta )\in L_{1}\text{ and }\underset{r\rightarrow +\infty }{\lim }\frac{%
\sigma \beta (\log r)}{\gamma (\beta (\log r))}=+\infty\\
\text{ for any }\sigma <\sigma _{(\alpha ,\beta )}[f]\text{;} \\
\sigma _{(\alpha ,\gamma (\beta ))}[f]=0~\ \ \text{when }\gamma (\beta )\in
L_{1}\text{ and }\underset{r\rightarrow +\infty }{\lim }\frac{\text{ }\sigma
_{1}\beta (\log r)}{\gamma (\beta (\log r))}=0\\
\text{ for any }\sigma_{1}>\sigma _{(\alpha ,\beta )}[f]\text{;} \\
\sigma _{(\gamma (\alpha ),\gamma (\beta ))}[f]=1~\ \ \text{when }\gamma \in
L_{si}.
\end{array}%
\right.
\end{equation*}
\end{definition}

\begin{remark}
\label{r1}An entire function $f(z)$ is said to have generalized index-pair $%
\left( \alpha ,\beta \right) $ if $0<\rho _{(\alpha ,\beta )}[f]<+\infty $
and $\rho _{\left( \exp \alpha ,\exp \beta \right) }\left[ f\right] $ is not
a nonzero finite number.
\end{remark}



\begin{proposition}
\label{p4}Let $f_{1}(z),$ $f_{2}(z)$ be non-constant meromorphic functions
with $\sigma _{(\alpha (\log ),\beta )}[f_{1}]$ and $\sigma _{(\alpha (\log
),\beta )}[f_{2}]$ as their $(\alpha (\log ),\beta )$-order. Then\newline
(i) $\sigma _{(\alpha (\log ),\beta )}[f_{1}\pm f_{2}]\leq \max \{\sigma
_{(\alpha (\log ),\beta )}[f_{1}],$ $\sigma _{(\alpha (\log ),\beta
)}[f_{2}]\}$;\newline
(ii) $\sigma _{(\alpha (\log ),\beta )}[f_{1}\cdot f_{2}]\leq \max \{\sigma
_{(\alpha (\log ),\beta )}[f_{1}],$ $\sigma _{(\alpha (\log ),\beta
)}[f_{2}]\}$;\newline
(iii) If $\sigma _{(\alpha (\log ),\beta )}[f_{1}]\neq \sigma _{(\alpha
(\log ),\beta )}[f_{2}]$, then
\begin{equation*}
\sigma _{(\alpha (\log ),\beta )}[f_{1}\pm f_{2}]=\max \{\sigma _{(\alpha
(\log ),\beta )}[f_{1}],\sigma _{(\alpha (\log ),\beta )}[f_{2}]\};
\end{equation*}%
(iv) If $\sigma _{(\alpha (\log ),\beta )}[f_{1}]\neq \sigma _{(\alpha (\log
),\beta )}[f_{2}]$, then
\begin{equation*}
\sigma _{(\alpha (\log ),\beta )}[f_{1}\cdot f_{2}]=\max \{\sigma _{(\alpha
(\log ),\beta )}[f_{1}],\sigma _{(\alpha (\log ),\beta )}[f_{2}]\}.
\end{equation*}
\end{proposition}

\begin{proof}
Without loss of generality, we assume that
\begin{equation*}
\sigma _{(\alpha (\log ),\beta )}[f_{1}]\leq \sigma _{(\alpha (\log ),\beta
)}[f_{2}]<+\infty.
\end{equation*}%
From the definition of $(\alpha (\log ),\beta )$-order, for any given $%
\varepsilon >0$, we obtain for all sufficiently large values of $r$ that%
\begin{equation}
T(r,f_{1})<\exp ^{[2]}(\alpha ^{-1}((\sigma _{(\alpha (\log ),\beta
)}[f_{1}]+\varepsilon )\beta (\log r)))  \label{2a}
\end{equation}%
and%
\begin{equation}
T(r,f_{2})<\exp ^{[2]}(\alpha ^{-1}((\sigma _{(\alpha (\log ),\beta
)}[f_{2}]+\varepsilon )\beta (\log r)))\text{.}  \label{3a}
\end{equation}%
Since $T(r,f_{1}\pm f_{2})\leq T(r,f_{1})+T(r,f_{2})+\log 2$ for all large $%
r $, we get from (\ref{2a}) and (\ref{3a}) for all sufficiently large values
of $r$ that%
\begin{eqnarray*}
T(r,f_{1}\pm f_{2}) &<&2\exp ^{[2]}(\alpha ^{-1}((\sigma _{(\alpha (\log
),\beta )}[f_{2}]+\varepsilon )\beta (\log r)))+\log 2 \\
i.e.,~T(r,f_{1}\pm f_{2}) &<&3\exp ^{[2]}(\alpha ^{-1}((\sigma _{(\alpha
(\log ),\beta )}[f_{2}]+\varepsilon )\beta (\log r))) \\
i.e.,~\frac{1}{3}T(r,f_{1}\pm f_{2}) &<&\exp ^{[2]}(\alpha ^{-1}((\sigma
_{(\alpha (\log ),\beta )}[f_{2}]+\varepsilon )\beta (\log r)))
\end{eqnarray*}%
\begin{eqnarray*}
i.e.,~\left( 1+o\left( 1\right) \right) \log ^{[2]}T(r,f_{1}\pm f_{2})
&<&\alpha ^{-1}((\sigma _{(\alpha (\log ),\beta )}[f_{2}]+\varepsilon )\beta
(\log r)) \\
i.e.,~\alpha ((1+o(1))\log ^{[2]}T(r,f_{1}\pm f_{2})) &<&(\sigma _{(\alpha
(\log ),\beta )}[f_{2}]+\varepsilon )\beta (\log r) \\
i.e.,~(1+o(1))\alpha (\log ^{[2]}T(r,f_{1}\pm f_{2})) &<&(\sigma _{(\alpha
(\log ),\beta )}[f_{2}]+\varepsilon )\beta (\log r),
\end{eqnarray*}%
which implies that%
\begin{equation*}
\underset{r\rightarrow +\infty }{\lim \sup }\frac{(1+o(1))\alpha (\log
^{[2]}T(r,f_{1}\pm f_{2}))}{\beta (\log r)}\leq \sigma _{(\alpha (\log
),\beta )}[f_{2}]+\varepsilon
\end{equation*}%
holds for any given $\varepsilon >0$. Hence%
\begin{equation}
\sigma _{(\alpha (\log ),\beta )}[f_{1}\pm f_{2}]\leq \max \{\sigma
_{(\alpha (\log ),\beta )}[f_{1}],\sigma _{(\alpha (\log ),\beta )}[f_{2}]\}%
\text{.}  \label{4a}
\end{equation}



\begin{thebibliography}{99}
\bibitem{4} S. Bank and I. Laine, \textbf{On the oscillation theory of $%
f^{\prime \prime }+Af=0$ where $A$ is entire}.  Trans. Amer. Math. Soc. 273 (1982), no. 1, 351--363.

\bibitem{b1} B. Bela\"{\i}di and S. Hamouda, \textbf{Orders of solutions of
an $n$-th order linear differential equation with entire coefficients}.
Electron. J. Differential Equations 2001, No. 61, 5 pp.

\bibitem{b2} B. Bela\"{\i}di, \textbf{Estimation of the hyper-order of
entire solutions of complex linear ordinary differential equations whose
coefficients are entire functions}. Electron. J. Qual. Theory Differ. Equ.
2002, no. 5, 8 pp.

\bibitem{b3} B. Bela\"{\i}di,  \textbf{Growth of solutions of certain
non-homogeneous linear differential equations with entire coefficients}.
JIPAM. J. Inequal. Pure Appl. Math. 5 (2004), no. 2, Article 40, 9 pp.

\bibitem{b4} B. Bela\"{\i}di,  \textbf{Growth of $\rho_{\varphi}- $
order solutions of linear differential equations with entire coefficients}.
PanAmer. Math. J. 27 (2017), no. 4, 26--42.

\bibitem{b5} B. Bela\"{\i}di,  \textbf{Fast growing solutions to linear
differential equations with entire coefficients having the same $\rho
_{\varphi}-$order}. J. Math. Appl. 42 (2019), 63--77.

\bibitem{b6} B. Bela\"{\i}di and T. Biswas, \textbf{Complex oscillation of a
second order linear differential equation with entire coefficients of $
(\alpha ,\beta )-$order}. Silesian J. Pure Appl. Math. vol. 11
(2021), 1--21.

\bibitem{b7} B. Bela\"{\i}di and T. Biswas, \textbf{Growth properties of
solutions of complex differential equations with entire coefficients of
finite $(\alpha ,\beta ,\gamma )-$order}. Electron. J. Differential
Equations, 2023, No. 27, 14 pp.

\bibitem{b} L. G. Bernal, \textbf{On growth $k-$order of solutions
of a complex homogeneous linear differential equation}. Proc. Amer. Math.
Soc. 101 (1987), no. 2, 317--322.

\bibitem{bou} R. Bouabdelli and B. Bela\"{\i}di, \textbf{Growth and complex
oscillation of linear differential equations with meromorphic coefficients
of $\left[ p,q\right]-\varphi$ order. }Int. J. Anal.
Appl. 6 (2014), No. 2, 178--194.

\bibitem{CY} Z. X. Chen and C. C. Yang, \textbf{Some further results on the
zeros and growths of entire solutions of second order linear differential
equations}. Kodai Math. J. 22 (1999), no. 2, 273--285.

\bibitem{CS} I. Chyzhykov and N. Semochko, \textbf{Fast growing entire
solutions of linear differential equations}. Math. Bull. Shevchenko Sci.
Soc. 13 (2016), 1--16.

\bibitem{g} G. G. Gundersen, E. M. Steinbart and S. Wang, \textbf{The
possible orders of solutions of linear differential equations with
polynomial coefficients}. Trans. Amer. Math. Soc. 350 (1998), no. 3,
1225--1247.

\bibitem{g1} G. G. Gundersen, \textbf{Estimates for the logarithmic
derivative of a meromorphic function, plus similar estimates}. J. London
Math. Soc. (2) 37 (1988), no. 1, 88--104.

\bibitem{1} W. K. Hayman, \textbf{Meromorphic functions}. Oxford
Mathematical Monographs, Clarendon Press, Oxford 1964.

\bibitem{17} W. K. Hayman, \textbf{The local growth of power series: a
survey of the Wiman-Valiron method}. Canad. Math. Bull. 17 (1974), no. 3,
317--358.

\bibitem{18} Y. Z. He and X. Z. Xiao, \textbf{Algebroid Functions and
Ordinary Differential Equations}. Science Press, Beijing (1988) (in Chinese).

\bibitem{20} J. Heittokangas, R. Korhonen and J. R\"{a}tty\"{a}, \textbf{
Growth estimates for solutions of linear complex differential equations}.
Ann. Acad. Sci. Fenn. Math. 29 (2004), no. 1, 233--246.

\bibitem{21} G. Jank and L. Volkmann, \textbf{Einf\"{u}hrung in die Theorie
der ganzen und meromorphen Funktionen mit Anwendungen auf
Differentialgleichungen}. Birkh\"{a}user Verlag, Basel, 1985.

\bibitem{8} O. P. Juneja,  G. P. Kapoor  and S. K. Bajpai, \textbf{On the $%
(p,q)-$order and lower $(p,q)-$order of an entire function}%
. J. Reine Angew. Math. 282 (1976), 53--67.

\bibitem{9} O. P. Juneja,  G. P. Kapoor  and S. K. Bajpai, \textbf{On the $%
(p,q)-$type and lower $(p,q)-$type of an entire function}.
J. Reine Angew. Math. 290 (1977), 180--190.

\bibitem{k} M. A. Kara and B. Bela\"{\i}di, \textbf{Growth of $\varphi-$order solutions of linear differential equations with meromorphic
coefficients on the complex plane}. Ural Math. J. 6 (2020), no. 1, 95--113.

\bibitem{13}  L. Kinnunen, \textbf{Linear differential equations with
solutions of finite iterated order}. Southeast Asian Bull. Math. 22 (1998),
no. 4, 385--405.

\bibitem{19} I. Laine, \textbf{Nevanlinna theory and complex differential
equations}. De Gruyter Studies in Mathematics, 15. Walter de Gruyter \& Co.,
Berlin, 1993.

\bibitem{10} L. M. Li  and  T. B. Cao,  \textbf{Solutions for linear
differential equations with meromorphic coefficients of $(p,q)-$order in the plane}. Electron. J. Differential Equations 2012, No. 195, 15
pp.

\bibitem{11} J. Liu, J. Tu  and L. Z. Shi,  \textbf{Linear differential
equations with entire coefficients of $[p,q]-$order in the complex
plane}. J. Math. Anal. Appl. 372 (2010), no. 1, 55--67.

\bibitem{l} S. G. Liu, J. Tu and H. Zhang,  \textbf{The growth and zeros of
linear differential equations with entire coefficients of $[p,q]-\varphi$ order}. J. Comput. Anal. Appl., 27 (2019), no. 4,
681--689.

\bibitem{MST} O. M. Mulyava, M. M. Sheremeta and Yu. S. Trukhan, \textbf{Properties of solutions of a heterogeneous differential equation of the
second order}. Carpathian Math. Publ. 11 (2019), no. 2, 379--398.

\bibitem{DS}  D. Sato, \textbf{On the rate of growth of entire functions of
fast growth}. Bull. Amer. Math. Soc. 69 (1963), 411--414.

\bibitem{STX} X. Shen, J. Tu and H. Y. Xu, \textbf{Complex oscillation of a
second-order linear differential equation with entire coefficients of $%
[p,q]-\varphi $  order}. Adv. Difference Equ. 2014,
2014:200, 14 pp.

\bibitem{MNS} M. N. Sheremeta, \textbf{Connection between the growth of the
maximum of the modulus of an entire function and the moduli of the
coefficients of its power series expansion}. Izv. Vyssh. Uchebn. Zaved.
Mat., 2 (1967), 100--108. (in Russian).

\bibitem{SB} A. P. Singh and M. S. Baloria, \textbf{On the maximum modulus
and maximum term of composition of entire functions}. Indian J. pure appl.
Math. 22 (1991), 1019--1026.

\bibitem{t1} J. Tu, Z. X. Chen and X. M. Zheng, \textbf{Growth of solutions
of complex differential equations with coefficients of finite iterated order}.
Electron. J. Differential Equations 2006, No. 54, 8 pp.

\bibitem{t2} J. Tu and  C. F. Yi, \textbf{On the growth of solutions of a
class of higher order linear differential equations with coefficients having
the same order}. J. Math. Anal. Appl. 340 (2008), no. 1, 487--497.

\bibitem{t3} J. Tu and  Z. X. Chen, \textbf{Growth of solutions of complex
differential equations with meromorphic coefficients of finite iterated order}. Southeast Asian Bull. Math. 33 (2009), no. 1, 153--164.

\bibitem{v} G. Valiron, \textbf{Lectures on the general theory of integral
functions}. translated by E. F. Collingwood, Chelsea Publishing Company, New
York, 1949.

\bibitem{15} H. Y. Xu  and J. Tu, \textbf{Oscillation of meromorphic solutions
to linear differential equations with coefficients of $[p,q]-$order}%
. Electron. J. Differential Equations 2014, No. 73, 14 pp.

\bibitem{3} C. C. Yang and H. X. Yi, \textbf{Uniqueness theory of meromorphic
functions}. Mathematics and its Applications, 557. Kluwer Academic
Publishers Group, Dordrecht, 2003.
\end{thebibliography}
\vspace{1cm}
\address{A. Popa:  Department of Mathematics, Faculty of Sciences, University of.., ROMANIA }
\newline
e-mail: apopa@ub.ro
ORCID 0000-0013-0189-4040
\newline

\end{document}