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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}



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\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}
