Superfluous vertical spacing

Question

When a new section starts, I am facing extra spacing.

\documentclass[10pt]{report}
\usepackage{mathptmx,amsmath,amssymb,bm}
\begin{document}
\begin{align*}
& \kappa(ru+m(1-r)v)\leq\,r\kappa(u)+m(1-r)\kappa(v)\\
\begin{split}
& \chi\big[\kappa(ru+m(1-r)v)\big]\leq\,\chi\big[r\kappa(u)\\
&+m(1-r)\kappa(v)\big]
\end{split}
\\[2ex]
\begin{split}
& \big(\chi o \kappa\big)\big(ru+m(1-r)v\big)\leq\,h(r) \chi \big[\kappa(u)\big]\\
&+mh(1-r)\varphi\big(\chi \big[\kappa(u)\big],\chi \big[\kappa(v)\big]\big)
\end{split}
\\[2ex]
\begin{split}
& \big(\chi o \kappa\big)\big(ru+m(1-r)v\big)\leq\,h(r)\big(\chi o \kappa\big)(u)\\
&+mh(1-r)\varphi\big((\chi o \kappa)(u),(\chi o \kappa)(v)\big).
\end{split}
\\[2ex]
\end{align*}
\section{Hermite-Hadamard type inequalities.}
The incoming theorems follow ideas from [1]\\
The follwing theorem gives boundedness of $\varphi_{(h,m)}-$convex function and will be use in theorem (4)
Consider $\kappa:I\rightarrow \mathbb{R}$ be $\varphi_{(h,m)}-$convex function, such that $\varphi$ is bounded from above on $\kappa(I)\times \kappa(I)$, then for  $p,q\in I$ with $p<q$, $\kappa$ is bounded on $[p,q]\subseteq I$.
\end{document}

Answer 1

Probably it is off-topic, but anyway ...

It is not clear, why you use split inside align+ environment when your equations can be fit nicely on page without it (even if you number them):

\documentclass[10pt]{report}
\usepackage{mathptmx,amsmath,amssymb,bm}
\usepackage{amssymb,mathtools}

\begin{document}
    \begin{align*}
\kappa(ru+m(1-r)v)
    &   \leq\,r\kappa(u) + m(1-r)\kappa(v)                  \\
\chi\big[\kappa(ru+m(1-r)v)\big]
    &   \leq\,\chi\big[r\kappa(u) + m(1-r)\kappa(v)\big]    \\
\big(\chi o \kappa\big)\big(ru+m(1-r)v\big)
    &   \leq\,h(r) \chi \big[\kappa(u)\big] + mh(1-r)\varphi\big(\chi\big[\kappa(u)\big],\chi \big[\kappa(v)\big]\big)                                \\
\big(\chi o \kappa\big)\big(ru+m(1-r)v\big)
    &   \leq\,h(r)\big(\chi o \kappa\big)(u)+ mh(1-r)\varphi\big((\chi o \kappa)(u),(\chi o \kappa)(v)\big).
\end{align*}
Numbered as subequations:
\begin{subequations}
\begin{align}
\kappa(ru+m(1-r)v)
    &   \leq\,r\kappa(u) + m(1-r)\kappa(v)                  \\
\chi\big[\kappa(ru+m(1-r)v)\big]
    &   \leq\,\chi\big[r\kappa(u) + m(1-r)\kappa(v)\big]    \\
\big(\chi o \kappa\big)\big(ru+m(1-r)v\big)
    &   \leq\,h(r) \chi \big[\kappa(u)\big] + mh(1-r)\varphi\big(\chi\big[\kappa(u)\big],\chi \big[\kappa(v)\big]\big)                                \\
\big(\chi o \kappa\big)\big(ru+m(1-r)v\big)
    &   \leq\,h(r)\big(\chi o \kappa\big)(u)+ mh(1-r)\varphi\big((\chi o \kappa)(u),(\chi o \kappa)(v)\big).
\end{align}
\end{subequations}

\section{Hermite-Hadamard type inequalities.}
The incoming theorems follow ideas from [1].

The follwing theorem gives boundedness of $\varphi_{(h,m)}-$convex function and will be use in theorem (4). Consider $\kappa:I\rightarrow \mathbb{R}$ be $\varphi_{(h,m)}$-convex function, such that $\varphi$ is bounded from above on $\kappa(I)\times \kappa(I)$, then for  $p,q\in I$ with $p<q$, $\kappa$ is bounded on $[p,q]\subseteq I$.
\end{document}

Answer 2

I see no reason for split.

A few points to note.

1. \big is unnecessary in most places and should be either \bigl or \bigr for opening and closing

2. 2ex is really too large

3. \chi o \kappa is wrong and should be \chi \circ \kappa

4. $\varphi_{(h,m)}-$convex is wrong and should be $\varphi_{(h,m)}$-convex; if you really want a dash instead of a hyphen, use $\varphi_{(h,m)}$--convex

5. mathptmx is an honored 25-year-old hack, but it's not really the best way to get Times-like fonts nowadays

6. The display might be aligned at the ≤ symbol or at the left, take your pick

7. Using explicit cross-references is not the best way to go, look for \label and \ref in any LaTeX guide

8. Don't use \\ in order to end paragraphs, instead leave a blank line

\documentclass[10pt]{report}
\usepackage{newtxtext,newtxmath}
\usepackage{amsmath}
%\usepackage{amssymb}% not with newtxmath
\usepackage{bm}

\begin{document}

The display with alignment at $\leq$
\begin{align*}
\kappa(ru+m(1-r)v)&\leq r\kappa(u)+m(1-r)\kappa(v)
\\[1ex]
\chi[\kappa(ru+m(1-r)v)]&\leq\chi[r\kappa(u)+m(1-r)\kappa(v)]
\\[1ex]
(\chi \circ \kappa)(ru+m(1-r)v)&\leq h(r) \chi [\kappa(u)]
  +mh(1-r)\varphi\bigl(\chi [\kappa(u)],\chi [\kappa(v)]\bigr)
\\[1ex]
(\chi \circ \kappa)(ru+m(1-r)v)&\leq h(r)(\chi \circ \kappa)(u)
  +mh(1-r)\varphi\bigl((\chi \circ \kappa)(u),(\chi \circ \kappa)(v)\bigr).
\end{align*}

The display with left alignment
\begin{align*}
& \kappa(ru+m(1-r)v)\leq r\kappa(u)+m(1-r)\kappa(v)
\\[1ex]
& \chi[\kappa(ru+m(1-r)v)]\leq\chi[r\kappa(u)+m(1-r)\kappa(v)]
\\[1ex]
& (\chi \circ \kappa)(ru+m(1-r)v)\leq h(r) \chi [\kappa(u)]
  +mh(1-r)\varphi\bigl(\chi [\kappa(u)],\chi [\kappa(v)]\bigr)
\\[1ex]
& (\chi \circ \kappa)(ru+m(1-r)v)\leq h(r)(\chi \circ \kappa)(u)
  +mh(1-r)\varphi\bigl((\chi \circ \kappa)(u),(\chi \circ \kappa)(v)\bigr).
\end{align*}

\section{Hermite-Hadamard type inequalities.}

The incoming theorems follow ideas from [1].

The following theorem gives boundedness of $\varphi_{(h,m)}$-convex function 
and will be used in theorem~(4).

Consider $\kappa:I\rightarrow \mathbb{R}$ be $\varphi_{(h,m)}$-convex
function, such that $\varphi$ is bounded from above on $\kappa(I)\times \kappa(I)$, 
then for  $p,q\in I$ with $p<q$, $\kappa$ is bounded on $[p,q]\subseteq I$.

\end{document}