1.Link is up and running
2.Home page available with search box
Steps to reproduce:
1.open the link for Bookmyshow.
2.type for something in search box
1.When the web page is minimized then search box also has to be there for searching.
1.Web page is minimized and search box is missing.
As you state it, this is not a bug: minimizing a window is *meant* to temporarily hide its contents, including (in the case of a Firefox browser window) both the search box on the toolbar and any search boxes in the web page itself.
If you meant something different that is a bug, please report it to Bookmyshow and/or your browser's bug tracker (not here - this is Freedesktop Bugzilla, not Mozilla Bugzilla), and include the full URL and a screenshot.
*** Bug 100538 has been marked as a duplicate of this bug. ***