I90 CONTROLLER REPORTED NEAR SEPARATION LOSS AT 6000 FT WHEN FAILING TO CORRECT A DEVELOPING CONFLICT BETWEEN A DEPARTURE AND ARRIVAL AIRCRAFT; LISTING FATIGUE AS CONTRIBUTORY.

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A PC12 DEPARTED HOU NBOUND; AND I CLBED HIM TO 13000 FT. I NOTICED HE WASN'T CLBING THAT GREAT; BUT THERE WEREN'T ANY CONFLICTS AT THE TIME. I WAS ONLY WORKING A FEW OTHER ACFT; AND I MADE A FEW COORDS ON THEM AND HAD TO LOOK UP APCH INFO ON ONE OF THEM. DURING THAT TIME; 2 ARR ACFT FOR IAH ARRIVING ON THE S DOWNWIND BECAME A FACTOR FOR THE PC12 THAT WAS CLBING OUT SLOWLY. I HADN'T WATCHED CLOSELY ENOUGH AS THE CONFLICT DEVELOPED; BUT THE FINAL CTLR CALLED ME ON THE LANDLINE TO SAY HIS ACFT HAD VISUAL SEPARATION WITH MINE. I WASN'T EVEN AWARE OF WHAT HE WAS TALKING ABOUT UNTIL HE CAME OVER THE OTHER ACFT AT 7000 FT; 1000 FT APART. I DON'T KNOW TO WHAT EXTENT THAT SEPARATION WAS LOST; BUT IF I ONLY HAD 1000 FT WHEN THE PC12 WAS OVER THE E145; SEPARATION HAD TO HAVE BEEN MUCH LESS AT THEIR CLOSEST PROX. THE OTHER CTLR SAID HE NOTICED MY PC12 WHEN HE WAS ONLY CLBING OUT OF 6300 FT AND STARTED CALLING TFC. ALTHOUGH I ONLY HAD SEVERAL ACFT ON FREQ; FATIGUE CONTRIBUTED TO MY LACK OF FOCUS; PREVENTING ME FROM NOTICING AND CORRECTING THESE CONFLICTING ACFT. I WASN'T VERY BUSY; AND UNDER NORMAL CIRCUMSTANCES; I SHOULD'VE AND WOULD'VE BEEN ABLE TO CTL THE SITUATION. ATC CTLR SCHEDULES AND FATIGUE SERIOUSLY NEED TO BE ADDRESSED!