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No. ESXi 5.5 cannot be managed by vSphere 4.1
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VMware's interoperability matrix is where you can find this information:<br />
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<a href="http://www.vmware.com/resources/compatibility/sim/interop_matrix.php" rel="ugc">http://www.vmware.com/resources/compatibility/sim/interop_matrix.php</a><br />
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In your case, in the solution section I entered 'VMware vCenter Server' Version '4.1' and under Platform/Solution VMware ESX/ESXi. <br />
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This will provide an easy to read matrix. A green check mark/tick mean compatible, none means not.
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Hi Experts,<br />
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I am running vSphere 4.1 managing ESXi 4.1. If I want to upgrade to ESXi 5.5 can I still manage it with vSphere 4.1?
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Hi Experts,

I am running vSphere 4.1 managing ESXi 4.1. If I want to upgrade to ESXi 5.5 can I still manage it with vSphere 4.1?

No. ESXi 5.5 cannot be managed by vSphere 4.1

VMware&#39;s interoperability matrix is where you can find this information:

http://www.vmware.com/resources/compatibility/sim/interop_matrix.php

In your case, in the solution section I entered &#39;VMware vCenter Server&#39; Version &#39;4.1&#39; and under Platform/Solution VMware ESX/ESXi. 

This will provide an easy to read matrix. A green check mark/tick mean compatible, none means not.

What ESXi versions can I manage with vSphere 4.1

VMware, a software company founded in 1998,  was one of the first commercially successful companies to offer x86 virtualization. The storage company EMC purchased VMware in 1994. Dell Technologies acquired EMC in 2016. VMware’s parent company is now Dell Technologies.

VMware has many software products that run on desktops, Microsoft Windows, Linux, and macOS, which allows the virtualizing of the x86 architecture.  Its enterprise software hypervisor for servers, VMware vSphere Hypervisor (ESXi), is a bare-metal hypervisor that runs directly on the server hardware and does not require an additional underlying operating system.