Hi, it is probably because you can also search in the address bar.

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but cannot in mozilla firefox, no suggestions in child page showed while searching on the address bar,.

Hi, you should receive suggestions in the address bar, so please go to Options (cog wheel) - in the 3-bar menu unless you've moved it, > Search tab, and make sure that 'Provide search suggestions' is ticked/checked. Also go to the Privacy tab (Location Bar), and tick/check your preferred options.

You can use the search bar on the about:home page to have the focus set to the search bar on the web page and not the location/address bar.

• Tools > Options > General > Startup: Home page = about:home