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I purchased an LED lighted display stand. When I take a photo with my iPhone the white stand is now lit in purple?

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Most natural light sources emit a broad spectrum of the wavelengths of visible light. The dominant color of that light is also usually very close to a range of colors emitted by "black body radiators" at various temperatures. Stars, for instance, are black body radiators. So are most metals when they are heated enough to glow. Regardless of the chemical makeup of many metals, they all glow at the same color when raised to the same temperature. Thus we define "color temperature" as the corresponding temperature, expressed on the Kelvin scale that begins at absolute zero (-273°C or -460°F), of black body radiators that radiate specific colors.

The Sun may be centered on about 5500 K, but there is a little bit of the entire visible spectrum in sunlight. That's generally because a black body radiator is not the exact same temperature over the entire surface that is emitting light. It's also because the Earth's atmosphere absorbs some of the energy radiated by our sun, and lets what we call visible light pass more easily. (Of course the reason we call it "visible light" is because we have evolved to be visually sensitive to the wavelengths of electromagnetic radiation that the atmosphere we evolved in allows to pass most easily!)

LEDs are not natural light sources. The light they emit is not usually broad spectrum. There are parts of the visible spectrum missing from the light they emit. The dominant color they emit is also usually not along the axis of color we call "color temperature." It may be more magnta or it may be more green. The green ←→ magenta axis is more or less orthogonal/perpendicular to the blue ←→ amber color temperature axis. In order to adjust for this nontypical light, we adjust what is known as the camera's "white balance". This adjustment can take place along the blue ←→ amber "color temperature" axis, the green ←→ magenta "tint" axis, or both.

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If your phone allows you to alter color temperature and white balance correction, play around with it until the light from your display stand looks more natural. Since your current results have a purple (blue + magenta) tint to them, I'd start by adjusting in the green and amber directions. Just don't forget to set it back to "auto" or "normal" when you are done! If your phone has no such control, but you can save images in "raw" format, do so and then adjust white balance using a raw convertor application. They all should allow you to adjust both "color temperature" and "tint".

Because the light that illuminated the photo is not fuller spectrum, you may also need to use a selective color tool, often called an HSL, HSV, or HSB tool (for Hue - Saturation - Luminance/Value/Brightness) that allows adjusting the hue, saturation, and brightness of different colors independently, to remove color casts that remain after white balance correction.

The type of lighting is different in the examples in this answer to How to edit photos shot in fluorescent light, but the methodology of correction is the same. Obviously the direction and amounts of the various color adjustments would need to be suited to the light illuminating the scene.

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Choose a manual white balance (WB) in your camera app.

Seemingly, your app's auto WB does not work all too well in this setup, so all you can do is to set it manually.

Composite of a "proper" WB (left) and the same picture with thrown-off WB (right).

Composite of a "proper" WB (left) and the same picture with thrown-off WB (right). "proper" = "most probably not 100% accurate, but to my personal liking".

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