Cool but ugly SQL

I'm told this can be done.  But I'm having a bugger of a time figuring this
out.  And it's starting to be crunch time.  My file layout:

SHH_LIB    Char         10
SHH_FILE   Char         10
SHH_MEMBER Char         10
SHH_LCDATE Date         10
SHH_LCTIME Time          8
SHH_HASH   Char         20

The file key is the first five fields, which makes the key unique.
Duplicate values for the first three key fields are allowed and expected.
My task is to select the Lib/File/Member with the highest timestamp (LCDATE
& LCTIME), omitting any duplicate Lib/File/Member with earlier timestamps.
I'm also selecting certain library and file values.  Here is my current SQL:

 Select * from SRCMBRHASH where
       shh_Lib = 'PRODSRCE' and shh_File LIKE 'Q%'
   and shh_LCDate = (Select Max(shh_LCDate) from SRCMBRHASH
     where shh_Lib = 'PRODSRCE' and shh_File LIKE 'Q%' )
  Order by shh_Lib, shh_File, shh_Member, shh_LCDate desc,
            shh_LCTime desc

This "works", as in no syntax errors, but the result set is way too small.
I would expect every unique Lib/File/Member to appear (approximately 100),
but I'm only getting three rows.

Problem 1: How do I select the MAX of two fields, LCDate and LCTime?  (I
know, should be a real timestamp field, but that's a fix for the next
version.)
Problem 2: I think I need to compare shh_Lib to shh_Lib, shh_File to
shh_File, and shh_Member to shh_Member from the two selects in the
subselect.  But how?

TIA,
Dan