Best way to refactor of more complicated TeX

Question

I need do refactoring ugly equations.

\documentclass{article}

\usepackage{amsmath}

\begin{document}

$
  \sum_{l=1}^{n}
  \frac{x_{k}}{\left(\overset{n}{\underset{l=1}{\sum}}\vert
      x_{l}\vert^{p}\right)^{\frac{1}{p}}}\
  \frac{y_{k}}{\left(\overset{n}{\underset{l=1}{\sum}}\vert
      y_{l}\vert^{q}\right)^{\frac{1}{q}}} \text{ } \leq
  \overset{n}{\underset{l=1}{\sum}} \frac{x_{i}^{p}}{p
      \sum_{j=1}^n (x_{i}^{p})^{\frac{1}{p} p}} + \frac{y_{i}}{(q
      \sum_{j=1}^n y_{i}^{q})^{\frac{1}{q}q}}= \frac{1}{p} +
    \frac{1}{q} = 1
$
\end{document}

What is your experience to create more flexibility TeX?

I consider using command to underscore \sum. What is your opinion about this solution?

Answer 1

The posted code is missing a \right so that it generates an error. Once that is fixed, the main stylistic error in the coding is using \underset and \overset rather than limits on the sum, and using textstyle rather than displaystyle for the expression.

I would use something more like the following although it is rather wide so perhaps use an amsmath multi-line display environment such as align rather than \[ \]

\documentclass[a4paper]{article}

\usepackage{amsmath}

\begin{document}

\[
  \sum_{l=1}^{n}
  \frac{x_{k}}{\left(\sum_{l=1}^{n}\lvert x_{l}\rvert^{p}\right)^{\frac{1}{p}}}
  \frac{y_{k}}{\left(\sum_{l=1}^{n}\lvert y_{l}\rvert^{q}\right)^{\frac{1}{q}}}
   \leq
  \left(\sum_{l=1}^{n} \frac{x_{i}^{p}}{p \sum_{j=1}^n (x_{i}^{p})^{\frac{1}{p} p}} +
                     \frac{y_{i}}{(q \sum_{j=1}^n y_{i}^{q})^{\frac{1}{q}q}}\right)
    = \frac{1}{p} +\frac{1}{q}
     = 1
\]
\end{document}

Answer 2

I also suggest that you define macros for often used expressions. Here is an adaptation of David Carlisle's answer using macros:

Below, I have defined two macros for the two expressions that are repeated. The only difference between them is one uses x/p and the other uses y/q. Of course, you should pick more meaningful names for the two macros.

Besides ensuring consistency, this makes the code a bit easier to read.

Code:

\documentclass[a4paper]{article}
\usepackage{amsmath}

\newcommand*{\AbsSumFraction}[2]{%
    % #1 = variable: x,y
    % #2 = exponent p, q
    \frac{#1_{k}}{\left(\sum_{l=1}^{n}\lvert #1_{l}\rvert^{#2}\right)^{\frac{1}{#2}}}%
}
\newcommand*{\SumFraction}[2]{%
    % #1 = variable: x,y
    % #2 = exponent p, q
    \frac{x_{i}^{p}}{p \sum_{j=1}^n (x_{i}^{p})^{\frac{1}{p} p}}
}

\begin{document}
\[
    \sum_{l=1}^{n} \AbsSumFraction{x}{p} \AbsSumFraction{y}{q}
    \le \left( \sum_{l=1}^{n}  \SumFraction{x}{p} + \SumFraction{y}{q} \right)
    = \frac{1}{p} + \frac{1}{q}
    = 1
\]
\end{document}