How to find minimum within group value greater than certain value in r

I have the following data frame:

  PATIENT_ID VALUE
  1          8
  1          16
  1          24 
  2          50 
  2          56
  3          2
  3          70

Now I want to find all PATIENT_IDs that have a minimum that is greater than 48. In this example, the minimum of Patient 1,2,3 are 8, 40 and 2 respectively. So, it should return only PATIENT_ID = 2, since that is the only PATIENT_ID that has a minimum greater than 48.

Allow for grouping with by:

 by(data = df, INDICES = df$PATIENT_ID, FUN = function(x){ifelse(min(x$VALUE) > 48, min(x$VALUE), FALSE)})

which gives:

df$PATIENT_ID: 1
[1] 0
------------------------------------------------------------------------------------------- 
df$PATIENT_ID: 2
[1] 50
------------------------------------------------------------------------------------------- 
df$PATIENT_ID: 3
[1] 0