The most important question right now, in relevance to whether you may have a claim or not, is if you have ever damaged the phone. This includes dropping it, getting it wet, etc. The damage in the picture above seems to be in the LED/LCD screen (can't remember which one the s4 has?), a hardware issue, not a software issue. If it is in fact a hardware issue caused by user neglect, you will have no claim as far as warranty, but may still be able to file a claim with your insurance company, credit card company, or renters insurance for cheaper than the repair may cost you.
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