Line-breaking (splitting) an inequality

Question

How can I split the following inequality across three lines?

\documentclass[12pt]{report}
\usepackage{mathtools}
\begin{document}
\begin{eqnarray}\label{hhi1}
\frac{1}{q-p}\int_{p}^{q}\kappa(u)du\leq min\left{\kappa(p)\int_{0}^{1} h(r)dr +m\varphi\bigg(\kappa(p),\kappa\Big(\frac{q}{m}\Big)\bigg)\int_{0}^{1} h(1-r)dr,\kappa(q)\int_{0}^{1} h(r)dr +m\varphi\bigg(\kappa(q),\kappa\Big(\frac{p}{m}\Big)\bigg)\int_{0}^{1} h(1-r)dr\right}.
\end{eqnarray}
\end{document}

Answer 1

Here's a solution that nests an aligned environment inside a multline environment. This allows us to draw the reader's eye to the structure of the inequality.

\documentclass[12pt]{report}
\usepackage{mathtools}
\begin{document}
\begin{multline}\label{hhi1}
\frac{1}{q-p}\int_{p}^{q}\kappa(u),du  \
\begin{aligned}[b]
\leq\min\biggl{ 
&\kappa(p)\int_{0}^{1} h(r),dr 
 +m\varphi\Bigl(\kappa(p),\kappa\Bigl(\frac{q}{m}\Bigr)\Bigr)
 \int_{0}^{1} h(1-r),dr ,, \
&\kappa(q)\int_{0}^{1} h(r),dr 
 +m\varphi\Bigl(\kappa(q),\kappa\Bigl(\frac{p}{m}\Bigr)\Bigr)
 \int_{0}^{1} h(1-r),dr \biggr} ,.
\end{aligned}
\end{multline}
\end{document}

Answer 2

One environment that breaks line is the split. It must be inside equation-like environment.

As you used \left\{ at the beginning, you shall add \right. before ending and breaking the line (\\). If you don't, you get a error message: Extra }, or forgotten \right. \end{split}. In consequence, as there must be a \right\} at the last line, there must be added a \left..

Also notice that \min is a math command to write the word "min" as math function while just typing min may lead to a misleading "m" times "i" times "n".

Ampersand (&) is used to define the tabulation. I organized the sums below the \leq symbol.

You may also considere using \mathrm{d} to the ds of derivatives. Check derivatives for more information.

\documentclass[12pt]{report}
\usepackage{amsmath}
\begin{document}
\begin{equation}\label{hhi1}
  \begin{split}
    \frac{1}{q-p}\int_{p}^{q}\kappa(u)du & \leq \min \left{ \kappa(p)\int_{0}^{1} h(r)dr \right. \
    & +m\varphi\bigg(\kappa(p),\kappa\Big(\frac{q}{m}\Big)\bigg)\int_{0}^{1} h(1-r)dr,\kappa(q)\int_{0}^{1} h(r)dr \
    & \left.+m\varphi\bigg(\kappa(q),\kappa\Big(\frac{p}{m}\Big)\bigg)\int_{0}^{1} h(1-r)dr \right}.
  \end{split}
\end{equation}
\end{document}

---

Edit

Following David Carisle's advice, I change the equation block to

\begin{equation}
  \begin{split}
    \frac{1}{q-p}\int_{p}^{q}\kappa(u)du & \leq \min \left\{ \kappa(p)\int_{0}^{1} h(r)dr \right.  \\
    & + m \varphi \left. \bigg(\kappa(p),\kappa\Big(\frac{q}{m}\Big)\bigg)\int_{0}^{1} h(1-r)dr,\kappa(q)\int_{0}^{1} h(r)dr \right. \\
    & + m \varphi \left. \bigg(\kappa(q),\kappa\Big(\frac{p}{m}\Big)\bigg)\int_{0}^{1} h(1-r)dr \right\}.
  \end{split}
\end{equation}

so that the spacing between +, m \varphi and the first ( is not lost.

Answer 3

A small variation of @mico answer:

• multlined (defined in mathtools package) is inserted in equation environment
• for fraction used in variables argument is used \mfrac (defined in the nccmath package):

\documentclass[12pt]{report}
\usepackage{nccmath, mathtools}
\begin{document}
\begin{equation}\label{hhi1}
\begin{multlined}[0.9\linewidth]
    \frac{1}{q-p}\int_p^q \kappa(u), du    \
    \leq    \min\biggl{
        \kappa(p)\int_{0}^{1} h(r), dr + 
        m \varphi\left(\kappa(p),\kappa\Bigl(\mfrac{q}{m}\Bigr)\right)
        \int_{0}^{1} h(1-r), dr,           \

            \kappa(q)\int_0^1 h(r),dr + m \varphi\left(\kappa(q),\kappa\Bigl(\mfrac{p}{m}\Bigr)\right)
                 \int_0^1 h(1-r), dr
                 \biggr}
\end{multlined}
\end{equation}
\end{document}

Answer 4

As you asked if is it right using align, I'm adding this alternative answer using it.

Following Mico's answer that highlights both parts of \min, and as align numbers each line, it would be nice to number each equation with some relevancy, not only a non-significant line, so I defined two generic variables A and B -- your functions with integrals -- and then the inequality is displayed using both A and B previously defined.

You may also label each line if needed.

Note: It is not necessary to align all &s. They aligned because my editor does it with a simple button press.

\documentclass[12pt]{report}
\usepackage{amsmath}
\begin{document}
\begin{align}
  A                                    & = \kappa(p)\int_{0}^{1} h(r)dr + m \varphi \bigg(\kappa(p), \kappa \Big(\frac{q}{m}\Big)\bigg)\int_{0}^{1} h(1-r)dr \\
  B                                    & = \kappa(q)\int_{0}^{1} h(r)dr + m \varphi \bigg(\kappa(q), \kappa \Big(\frac{p}{m}\Big)\bigg)\int_{0}^{1} h(1-r)dr \\
  \frac{1}{q-p}\int_{p}^{q}\kappa(u)du & \leq \min \left\{A, B \right\}
\end{align}
\end{document}