Please make sure to following the guidelines described in How to make good reproducible pandas examples when asking pandas related questions, it helps a lot for reproducibility.
The key element for the answer is the pairing of the groupby and rolling methods. groupby will group all the records with the same clientid and rolling will select the correct amount of records for the mean calculation.
import pandas as pd
import numpy as np
# setting up the dataframe
data = [
['day 1', 2, 50],
['day 2', 2, 60],
['day 3', 1, 45],
['day 4', 2, 45],
['day 5', 3, 90],
['day 6', 3, 10],
['day 7', 2, 10]
]
columns = ['date', 'clientid', 'amounts']
df = pd.DataFrame(data=data, columns=columns)
rolling_mean = df.groupby('clientid').rolling(2)['amounts'].mean()
rolling_mean.index = rolling_mean.index.get_level_values(1)
df['client_rolling_mean'] = rolling_mean