Cessna Citation Undifferentiated or Other Model
Air Traffic Control Fully Certified
Air Traffic Control Time Certified In Pos 1 (yrs) 4.0
Situational Awareness
ATC Issue All Types; Conflict Airborne Conflict; Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy
Air Traffic Control Separated Traffic; Air Traffic Control Issued New Clearance
Procedure; Company Policy; Human Factors; Airspace Structure
Approach handed off to me; two aircraft both heading on the same path. Aircraft X was a slower Citation; and wanted lower altitude. Aircraft Y was much faster and wanted 35000 Feet for their final altitude. Aircraft X was in front and they shipped the aircraft to me while he was still well in their airspace at 10000 feet (they own up to 14000 feet) and climbing into my airspace at 17 feet. After the aircraft checked on; I quickly recognized the situation; that this slow flying and slow climbing aircraft would be in the way of the Aircraft Y; and likely would be stuck underneath if I climbed the Aircraft X. I called Departure Control and I asked if I stopped Aircraft X at 15000 feet; could you jump him with Aircraft Y? The controller said sure; no problem. I said okay; and let Departure climb Aircraft Y to 23000 feet; she acknowledged and we hung up. I noticed they vectored Aircraft Y eastbound in attempt to get above the Citation before turning him back on course. A couple of minutes later I notice Aircraft Y is now heading toward the Citation with over a 100 knot overtake and similar altitudes. My Supervisor stands behind me and notices Conflict Alert going off and asks what I am doing. I told him; Departure Control is still talking to Aircraft Y. They are using TRACON separation rules to get above the Citation that I am talking to.The aircraft now are less than five miles (because they were both inside the halo on Aircraft X) while Aircraft X was indicating 14100 feet and Aircraft Y was indicating 14400 feet. Aircraft Y then checks on with only three hundred feet of separation and less than five miles. At this point the only thing I can do is call the traffic and ask Aircraft Y to expedite climb through 16000 feet; so I can ensure separation. Once I achieved separation; I continued to climb Aircraft X reference the vacated altitudes of Aircraft Y and handed off to the next controller.There is nothing I could have done to prevent this from happening again. I was trying to give the best service to these two aircraft; safe and expeditious for each; but Departure Control did not ensure separation between these two like I had instructed them to.
ZLA ARTCC Controller reported a conflict between two aircraft on same path but different speeds.  Controller stated coordination with Departure Control contributed to the event.