There is an old movie that I saw when I was young where young, female cello player meets an older gentleman and eventually falls in love with him. He is a bit of a player and in the beginning of the movie, there was some scandal involving him and stewardesses on his flight. He and the cello player meet somehow -perhaps at a symphony?- but her father doesn't necessarily approve of the match, either because she is too young or because he wants her to focus on her cello playing. By the end of the movie, however, they fall in love and probably get married.

I don't know the title of this movie, but would like to know. Can anybody help?

I think the movie that you're talking about is called "Love in the Afternoon." It stars Audrey Hepburn and Gary Cooper. If this the correct movie, you should also watch Audrey Hepburn in "Breakfast at Tiffany's" and "Roman Holiday", they are great as well.