It's also possible to style the entire DataFrame in one go by passing axis=None to apply.
We can identify rows which have differing values in the specified columns by comparing the first column (column 0) with the remaining columns (column 1-2) and identifying where there are unequal values using ne on axis=0.
df[['prod', 'stage']].ne(df['pre-prod'], axis=0)
# prod stage
# 0 False False
# 1 False True
Then we can check across rows for any rows which have any True values (meaning there is something that's not equal in the row).
df[['prod', 'stage']].ne(df['pre-prod'], axis=0).any(axis=1)
# 0 False
# 1 True
# dtype: bool
We can then simply apply the styles anywhere there's a True value in the resulting Series.
Altogether this could look something like:
def colour_rows_that_dont_match(df_: pd.DataFrame, comparison_cols: List[str]):
# Sanity check that comparison_cols is what we expect
assert isinstance(comparison_cols, list) and len(comparison_cols) > 1, \
'Must be a list and provide at least 2 column to compare'
# Create an Empty DataFrame to hold styles of the same shape as the original df
styles_df = pd.DataFrame('', index=df_.index, columns=df_.columns)
# Compare the first column's (col 0) values to the remaining columns.
# Find rows where any values are not equal (ne)
rows_that_dont_match = df[comparison_cols[1:]].ne(df[comparison_cols[0]], axis=0).any(axis=1)
# Apply styles to rows which meet the above criteria
styles_df.loc[rows_that_dont_match, :] = 'background-color: red'
return styles_df
df.style.apply(
colour_rows_that_dont_match,
# This gets passed to the function
comparison_cols=['pre-prod', 'prod', 'stage'],
# Apply to the entire DataFrame at once
axis=None
).to_html(buf='test_df.html')
Which produces the following:

Setup, version, and imports:
from typing import List
import pandas as pd # version 1.5.2
df = pd.DataFrame({
'App_name': ['matching-image', 'mismatching-image'],
'pre-prod': ['nginx', 'nginx'],
'prod': ['nginx', 'nginx'],
'stage': ['nginx', 'nginx:1.23.3-alpine']
})