Lookup or calculate nearest bigger value of table items where items are not unique.

Dear All Excel MASTERS, HLEP!

Let's say I have an excel file which have 2 sheets : "Inbox" and "Sent"
in those sheets, there are 2 columns : "Name" and "SCORE"
what I am trying to do, is to always get the nearest bigger "Score" (minimum value which is bigger) for any "Name" in the "Inbox" table, from the "Sent" Table. Please think of sheets bellow :

Inbox Table :

Sent table :

the goal is to get result like the bellow :

 
I tried many formulas for this, as you can see in bellow screenshot, after hours! I managed to find out I NEED HELP! then if you Excel masters help me solve this, I really would appreciate it.

the closest formula I have used is as bellow :

{=IF(Inbox!A2=Sent!$A$1:$A$8,MIN(IF(Sent!$B$1:$B$8>Inbox!B2,Sent!$B$1:$B$8)))}

screenshot, lot's of try! no luck. only the first two columns are data! others are my failures!