When you turn off the device without setting the volume to “silent” it will play a short melody together with the Samsung boot animation when turning it on again (on some other devices this will occur always independent of any volume setting).
If the device is rooted you can just rename the relevant OGG file to disable this.
I don’t take any responsibility to potential damages or failure caused by modifications of system files! Any modification at your own risk!
The file is
PowerOn.ogg in the path
/system/media/audio/ui. If you rename this to
PowerOn.ogg.disabled (for example with Root Browser or TotalCommander) the S3 Mini will be silent when turning it on.
By the way: With the same method you can also disable other sounds as for example the shutter sound of the camera.