我正在用期刊的 Latex 格式撰写研究论文，并且在“$\varphi_{m}$-凸函数的一些属性”部分前后遇到了多余的空格。请指出问题！
\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{[email protected]}
\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}
可以找到该类这里。
答案1
首先，这个类似乎被破坏了。它需要
\PassOptionsToPackage{dvipsnames}{color}
之前，否则类文件第 573 行\documentclass使用的颜色是未定义的。Green
但是，这样一来，该部分前后就没有太多空白了，只有前面有 21dd，下面有 10.5dd。如果要更改此设置，则必须复制：
\def\section{\@startsection{section}{1}{\z@}%
{-21dd plus-4pt minus-4pt}{10.5dd plus 4pt
minus4pt}{\large\bfseries}}
从类文件的第 1624-1626 行放入文档前言中，添加\makeatletter前后\makeatother内容并更改值，例如使用：
\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}
这将导致
然而，这样的改变通常不受自设课程的期刊出版商欢迎。所以你不应该这么做。
但也许你的代码并没有真正显示出你的问题。也许它只发生在分栏处？在这种情况下，像盒子这样不可破坏的材料可能会导致垂直空白过度拉伸。在这种情况下，你会看到underfull \vbox消息在日志文件中。例如，您可以添加\newpage命令以强制分页而不拉伸或使用\raggedbottom。但\raggedbottom如果不中断任何内容，也会停止最后一行的垂直对齐。
您可以使用此示例来试验替代方案：
\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}
第一张图片显示的是没有使用 的结果\raggedbottom，第二张图片显示的是使用 的结果\raggedbottom：
但是，由于出版商通常会进行最终编辑，因此不需要进行这样的更改，并且\raggedbottom通常不允许使用。