extra vertical spacing when new section starts in my research article, while working in the format of journal

Question

I am writing a research paper in a latex format of a journal, and facing extra spacing before and after of section ``Some properties for $\varphi_{m}$-convex function". point out the problem please!

    \documentclass[10pt]{NSP1}
\usepackage{url,floatflt}
\usepackage{helvet,times}
\usepackage{psfig,graphics}
\usepackage{mathptmx,amsmath,amssymb,bm}
\usepackage{float}
\usepackage[bf,hypcap]{caption}
\usepackage{xcolor}
\usepackage{bigints}
\usepackage{enumitem}
\usepackage[utf8]{inputenc}
\definecolor{rulecolor}{rgb}{0.0,0.54,0.44}
\tolerance=1
\emergencystretch=\maxdimen
\hyphenpenalty=10000
\hbadness=10000
\topmargin=0.00cm
\def\sm{\smallskip}
\def\no{\noindent}
\def\firstpage{1}
\setcounter{page}{\firstpage}
\def\thevol{7}
\def\thenumber{?}
\def\theyear{2022}
\DeclareUnicodeCharacter{2212}{-}
\begin{document}
\titlefigurecaption{{\large \bf \rm Applied Mathematics & Information Sciences }\ {\it\small An International Journal}}
\title{On $\varphi_{m}-$Convex Functions.}
\author{Migwel Vivas Cortez\hyperlink{author1}{$^1$}, Muhammad Shoib Saleem\hyperlink{author2}{$^2$} and Razi Ur Rehman\hyperlink{author3}{$^3$}}
\institute{$^1$Departamento de Matem'{a}ticas, Universidad Centroccidental Lisandro Alvarado, Barquisimeto, Venezuela\
$^2$Department Of Mathematics, University Of Okara, Okara, Punjab, Pakistan\
$^3$Department Of Mathematics, University Of Okara, Okara, Punjab, Pakistan}
\titlerunning{On $\varphi_{m}-$Convex Functions.}
\authorrunning{M. vivas, M. Shoib, R. Rehman}
%corresponding author email
\mail{razi.ur.rehman2089@gmail.com}
\received{...}
\revised{...}
\accepted{...}
\published{...}
\abstracttext{We will A for the convex function.. We give some basic properties for this notion. Furthermore, we set down proofs of Hermite-Hadamard type and Hermite-Hadamard-Fej'{e}r type integral inequalities for this notion.
}
\keywords{Convex function, $\varphi-$convex functions, A, B, , Hermite-Hadamard type inequalities and Hermite-Hadamard-Fej'{e}r type integral inequalities.}
\maketitle
\section{Introduction}
Through out the paper, we will use the symbol $``\kappa"$ for convex function (and its generalizations). Let $\varphi:\mathbb{R}\times \mathbb{R}\rightarrow\mathbb{R}$ be a function of two real variables unless we shall specify otherwise.
In the present section, we give some basic definitions and inequalities, which already exist in the literature, we use them through the paper. In section \ref{sec2}, we investigate some a \cite{6}. By using one of the four forms and \cite{6}, we introduce our new notion C, which is generalizations of convex, A and B. Let we shall see later. The remaining sections are clear by their title.
The following definition \cite{1}, \cite{2}, is the base of the literature:\
$\kappa:A\subset\mathbb{R}\rightarrow\mathbb{R}$ is known as \textbf{\textit{convex function}} if,
\begin{equation}
\kappa(ru+(1-r)v)\leq,r \kappa(u)+(1-r) \kappa(v)
\end{equation}
for every $u,v\in A,, and,,r\in[0,1].$
An inequality \cite{3}, \cite{4}, which is very basic and fundamental for the literature:\
If $\kappa:A\subset\mathbb{R}\rightarrow\mathbb{R}$ is convex function and $p,q\in A$ with $p<q$. Then
\begin{equation}\label{hhi}
\kappa\bigg(\frac{p+q}{2}\bigg)\leq\frac{1}{q-p}\int_{p}^{q}\kappa(u),\mathrm{d}u\leq \frac{\kappa(p)+\kappa(q)}{2}
\end{equation}
is called \textbf{\textit{Hermite-Hadamard inequality}}.
Another inequality \cite{5}, which is the generalization of above inequality (\ref{hhi}) was derived in the year 1905 Leopold Fej'{e}r, as the following:\
If $\kappa:[p,q]\subset\mathbb{R}\rightarrow\mathbb{R}$ is a convex function, and $\chi:[p,q]\rightarrow\mathbb{R}$ is symmetric about $\frac{p+q}{2}$, integrable and non-negative. Then
\begin{align}\label{hhfi}
\kappa\bigg(\frac{p+q}{2}\bigg)\int_{p}^{q}\chi(u),\mathrm{d}u\leq\frac{1}{q-p}\int_{p}^{q} \kappa(u) \chi(u),\mathrm{d}u\leq \notag\
\frac{\kappa(p)+\kappa(q)}{2}\int_{p}^{q} \chi(u),\mathrm{d}u
\end{align}
is known as \textbf{\textit{Hermite-Hadamard-Fej'{e}r inequality}}.
G. Toader \cite{6} , generalize the convex function as $m-$convex function, in the year 1984, as the following:\
$\kappa:[0,q)\subset\mathbb{R}\rightarrow\mathbb{R}$, $q>0$ be an \textbf{\textit{$m-$convex function}} if,
\begin{eqnarray}\label{mcon}
\kappa(ru+m(1-r)v)\leq r \kappa(u)+m(1-r) \kappa(v)
\end{eqnarray}
holds for every $u,v\in[0,q)$ and $r,m\in[0,1]$.
M. Eshaghi Gordji, M. Rostamian Delavar, M. De La Sen \cite{7}, generalize convex function as $\varphi-$convex function,in the year 2016,  as the following:\
let $A\subseteq\mathbb{R}$ and $\varphi:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ be a function of two real variables then, a function $\kappa:A\rightarrow\mathbb{R}$ is called \textbf{\textit{$\varphi-$convex}} if,
\begin{eqnarray}\label{phicons}
\kappa(ru+(1-r)v)\leq,\kappa(v)+r\varphi\Big(\kappa(u),\kappa(v)\Big).
\end{eqnarray}
Note that $\eta-$convex function \cite{10} and $\varphi-$convex function \cite{8} are the same notions. So we can also termed $\varphi_{m}-$convex function as $\eta_{m}-$convex function 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  sec 2   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Basic Definitions.}\label{sec2}
In this section we investigate different  forms of $\varphi-$convex function. One of these forms is require for our new generalization . We give some important remarks and examples for our new notion $\varphi_m-$convex function.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% def 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Now we give following four forms of $\varphi-$convex function, or it can be define in the following four different ways.
\begin{definition} \label{phicon}
Let $A\subseteq\mathbb{R}$, then a function $\kappa:A\rightarrow\mathbb{R}$ is called \textbf{\textit{$\varphi-$convex}} if, 
\begin{align}
& \kappa(ru+(1-r)v)\leq,r \label{phicon1} \ 
& \kappa(ru+(1-r)v)\leq,(1-r) \label{phicon2}\
& \kappa(ru+(1-r)v)\leq,\kappa  \label{phicon3}\
&\kappa(ru+(1-r)v)\leq,\kappa \label{phicon4}
\end{align}
for every $r\in[0,1],$ and, for every $u,v\in A$. Above inequality (\ref{phicon3}) is same as inequality (\ref{phicons}).
\end{definition}
The above definitions will become classical convex functions if, we take\
$\varphi(u,v)=v$ in (\ref{phicon1})\
$\varphi(u,v)=u$ in (\ref{phicon2})\
$\varphi(u,v)=u-v$ in (\ref{phicon3})\
$\varphi(u,v)=v-u$ in (\ref{phicon4}).
%%%%%%%%%%%%%%%% rem 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{remark}
All the four definitions are similar to each other. Lets see! if we set K in inequality (\ref{phicon1}), we get inequality (\ref{phicon4}) and if we set G in inequality (\ref{phicon2}), we get inequality (\ref{phicon3}), where H is another function of two real variables.
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% def 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition}
If we take equalities in-place of inequalities in Definition \ref{phicon}, we get \textbf{\textit{$\varphi-$affine functions}}, for all $r,u,v\in\mathbb{R}$. Clearly we will also get classical affine functions.
\end{definition}
We give one example to illustrate $\varphi-$convex function \ref{phicon1}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Example 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{example}
Let $\kappa(u)=u^2$ which is convex. If $\varphi(u,v)=2v+u$, then $\kappa$ is $\varphi-$ convex.
\end{example}
\textbf{Solution}.
\begin{align}
 \kappa(ru+(1-r)v)\
& =(ru+(1-r)v)^2\
& =r^2\
& \leq ru)\
& = ru)\ 
&= r
\end{align}
Which shows $\kappa$ is $\varphi-$convex.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% def 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Now we give different forms of $\varphi-$quasi convex function.
\begin{definition} \label{phiqcon}
Let $A\subseteq\mathbb{R}$, then a function $\kappa:A\rightarrow\mathbb{R}$ is called \textbf{\textit{$\varphi-$quasi convex}} if, 
\begin{align}
& \kappa(ru+(1-r)v)\leq\ A \label{phiqcon1}\
& \kappa(ru+(1-r)v)\leq A\label{phiqcon2}\
& \kappa(ru+(1-r)v)\leq A          \label{phiqcon3}\
&\kappa(ru+(1-r)v)A  \label{phiqcon4}
\end{align}
for every $r\in[0,1],$ and, for every $u,v\in A$.
\end{definition}
The above definitions will become classical quasi convex function if, we take\
$\varphi(u,v)=v$ in \ref{phiqcon1}\
$\varphi(u,v)=u$ in \ref{phiqcon2}\
$\varphi(u,v)=u-v$ in \ref{phiqcon3}\
$\varphi(u,v)=v-u$ in \ref{phiqcon4}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% def 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition}
If we reverse the inequalities in Definition \ref{phicon} and Definition \ref{phiqcon} then, we get \textbf{\textit{$\varphi-$concave}} and \textbf{\textit{$\varphi-$quasi concave}} functions.
\end{definition}
Through the rest of this paper, let $[0,q]=I\subset\mathbb{R}, q>0, [0,+\infty)=J\subset\mathbb{R}$ and $m,r\in[0,1]$, unless we specify otherwise.
Now we come to our main concern and construct A.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% def 5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition}\label{phihm}
$\kappa:I\rightarrow\mathbb{R}$ is \textbf{\textit{convex function}} with respect to non-negative $\varphi$  if, 
\begin{align}
\begin{split}
\kappa(r\
&r\kappa(u)
\end{split}
\end{align}
for every $u,v\in I$ and for every $r\in(0,1)$.
\end{definition}
we are denoting the set of all A as a class V.
If we choose K , we come to B \ref{mcon}.
If we choose K, we come to C \ref{phicon} (actually $\varphi-$convexity \cite{8}) for the interval $I$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% def 6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition}
If we reverse the inequality in Definition \ref{phihm}, then we get \textbf{\textit{concave function}}.
\end{definition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Example 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We give one example to illustrate our $\varphi_{m}-$convex function.
\begin{example}
let $\kappa(u)=u^2$ which is convex. If $\varphi(u,v)=2v+u$ then, $\kappa$ is $\varphi_{m}-$ convex.
\end{example}
\textbf{Solution}.
\begin{align}
\kappa(ru+m(1-r)v)\
& =(ru\
& =r^2\
& \leq r\
& = r\
&= r.
\end{align}
Which shows $\kappa$ is $\varphi_{m}-$ convex.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% section 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\section{Operations Which Preserve $\varphi_{m}$-Convex Function.}
In the present section, we shall give some basic properties for our notion $\varphi_{m}-$ convex function.  we first give various conditions for the function $\varphi$. We use these concepts often in our results.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% def 9 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition}
We say that $\varphi$ is,
\begin{enumerate}[label=(\roman*)]
\item\label{2} additive if, $\varphi(u_{1},v_{1})+\varphi(u_{2},v_{2})=\varphi(u_{1}+u_{2},v_{1}+v_{2})$ for all $u_{1},u_{2},v_{1},v_{2}\in\mathbb{R}.$\
\item\label{1} non-negatively homogeneous if, $\varphi(\beta u,\beta v)=\beta\varphi(u,v)$ for all $u,v\in\mathbb{R}$ and $\beta\geq 0.$\
\item no-negatively linear if, it satisfies conditions \ref{1} and \ref{2}.
\end{enumerate}
\end{definition}
The following is trivial fact of calculus for function of two variables:
Let limit of $u_n$ and $v_n$ exists in $\mathbb{R}$ and $f:\mathbb{R}^2\rightarrow\mathbb{R}^2$ is continuous, then
$$\lim_{n\to\infty}f(u_n,v_n)=f(\lim_{n\to\infty}u_n,\lim_{n\to\infty}v_n).$$
\end{document}

The class can be found here.

Answer 1

First of all, the class seems to be broken. It needs

\PassOptionsToPackage{dvipsnames}{color}

before \documentclass, otherwise the used color Green in line 573 of the class file, is undefined.

However with this, there isn't a lot of white space before and after the section, only 21dd before and 10.5dd below. If you want to change this you have to copy:

\def\section{\@startsection{section}{1}{\z@}%
    {-21dd plus-4pt minus-4pt}{10.5dd plus 4pt
     minus4pt}{\large\bfseries}}

from line 1624–1626 of the class file into your document preamble, add \makeatletter before and \makeatother after and change the values, e.g., using:

\PassOptionsToPackage{dvipsnames}{color}
\documentclass[10pt]{NSP1}
\usepackage{url,floatflt}
\usepackage{helvet,times}
\usepackage{psfig,graphics}
\usepackage{mathptmx,amsmath,amssymb,bm}
\usepackage{float}
\usepackage[bf,hypcap]{caption}
\usepackage{xcolor}
\usepackage{bigints}
\usepackage{enumitem}
\usepackage[utf8]{inputenc}
\usepackage{cite}
\definecolor{rulecolor}{rgb}{0.0,0.54,0.44}
\tolerance=1
\emergencystretch=\maxdimen
\hyphenpenalty=10000
\hbadness=10000
\topmargin=0.00cm
\def\sm{\smallskip}
\def\no{\noindent}
\def\firstpage{1}
\setcounter{page}{\firstpage}
\def\thevol{16}
\def\thenumber{6}
\def\theyear{2022}
\DeclareUnicodeCharacter{2212}{-}
\makeatletter
\def\section{@startsection{section}{1}{\z@}%
    {-10dd plus-4pt minus-4pt}{5dd plus 4pt
     minus4pt}{\large\bfseries}}
\makeatother
\begin{document}
    \begin{example}
    let $\kappa(u)=u^2$ which is convex. If $\varphi(u,v)=2v+u$ then, $\kappa$ is $\varphi_{m}-$ convex.
    \end{example}
    \textbf{Solution}.
    \begin{align}
    \kappa(ru+m(1-r)v)\
    & =(ru+m(1-r)v)^2\
    & =r^2u^2+m^2(1-r)^2v^2+rm(1-r)2uv\
    & \leq ru^2+m(1-r)v^2+m(1-r)(u^2+v^2)\
    & = ru^2+m(1-r)(u^2+2v^2)\
    &= r\kappa(u)+m(1-r)\varphi\Big(\kappa(u),\kappa(v)\Big).
    \end{align}
    Which shows $\kappa$ is $\varphi_{m}-$ convex.
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% section 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
    \section{Some properties for $\varphi_{m}$-convex function.}
In the present section, we shall give some basic properties for our notion $\varphi_{m}-$ convex function.  we first give various conditions for the function $\varphi$. We use these concepts often in our results.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% def 9 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition}
We say that $\varphi$ is,

\end{definition}

\end{document}

This would result in

However, such changes are usually not welcome by publishers of journals, that provide their own classes. So you should not do it.

But maybe your code doesn't really show your problem. Maybe it occurs only with column breaks? In this case, unbreakable material like boxes could result in overstretching vertical white spaces. In this case, you will see underfull \vbox messages in the log-file. You can, e.g., either add, \newpage commands to force page breaks without stretching or use \raggedbottom. But \raggedbottom would also stop vertical alignment of the last line, if it does not break anything.

You can use this example to experiment with the alternatives:

\PassOptionsToPackage{dvipsnames}{color}
\documentclass[10pt]{NSP1}
\usepackage{url,floatflt}
\usepackage{helvet,times}
\usepackage{psfig,graphics}
\usepackage{mathptmx,amsmath,amssymb,bm}
\usepackage{float}
\usepackage[bf,hypcap]{caption}
\usepackage{xcolor}
\usepackage{bigints}
\usepackage{enumitem}
\usepackage[utf8]{inputenc}
\usepackage{cite}
\definecolor{rulecolor}{rgb}{0.0,0.54,0.44}
\tolerance=1
\emergencystretch=\maxdimen
\hyphenpenalty=10000
\hbadness=10000
\topmargin=0.00cm
\def\sm{\smallskip}
\def\no{\noindent}
\def\firstpage{1}
\setcounter{page}{\firstpage}
\def\thevol{16}
\def\thenumber{6}
\def\theyear{2022}
\DeclareUnicodeCharacter{2212}{-}
\usepackage{lipsum}
\begin{document}
\raggedbottom% try without it to compare
    \begin{example}
    let $\kappa(u)=u^2$ which is convex. If $\varphi(u,v)=2v+u$ then, $\kappa$ is $\varphi_{m}-$ convex.
    \end{example}
    \textbf{Solution}.
    \begin{align}
    \kappa(ru+m(1-r)v)\
    & =(ru+m(1-r)v)^2\
    & =r^2u^2+m^2(1-r)^2v^2+rm(1-r)2uv\
    & \leq ru^2+m(1-r)v^2+m(1-r)(u^2+v^2)\
    & = ru^2+m(1-r)(u^2+2v^2)\
    &= r\kappa(u)+m(1-r)\varphi\Big(\kappa(u),\kappa(v)\Big).
    \end{align}
    Which shows $\kappa$ is $\varphi_{m}-$ convex.
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% section 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
    \section{Some properties for $\varphi_{m}$-convex function.}
In the present section, we shall give some basic properties for our notion $\varphi_{m}-$ convex function.  we first give various conditions for the function $\varphi$. We use these concepts often in our results.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% def 9 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition}
We say that $\varphi$ is,


\end{definition}
\lipsum[1-2]
  % \newpage% This would be an alternative to \raggedbottom above.
  \rule{8cm}{5cm}
\end{document}

Here the first image shows the result without \raggedbottom and the second one the result with \raggedbottom:

However, because usually the publisher will do the final editing, such changes are not needed and using \raggedbottom is usually not allowed.